Abstract
This chapter starts with a survey of evaluations of exponential sums, and then brings results dealing with the zeta-function and prime number theory. In particular results on differences of consecutive primes are discussed. Certain multiplicative problems, including questions of the existence of consecutive power residues and prime divisors of polynomial values are then presented, and the section on analytic methods ends with information about the circle method, its various applications (problems of Waring, Waring–Kamke and Hilbert–Kamke, …) and the conjectures of Hardy and Littlewood dealing with additive prime number theory. It follows a description of the beginnings of the class-field theory and the related work of Artin as well as the discovery by Hasse of the local-global principle. The main achievements in geometry of numbers and Diophantine approximations of that period include Siegel’s strengthening of Thue’s theorem and Khintchine’s application of measure theory in approximation theory. The chapter ends with a section on Diophantine equations (Siegel’s method and the beginning of modern theory of elliptic curves).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Colin Maclaurin (1698–1746), professor in Aberdeen and Edinburgh. See [6235].
- 2.
- 3.
It was later proved by V. Jarník and E. Landau [3116] that the optimal value of this constant equals 1/2+1/π+(1/4+1/π 2)1/2=1.4110….
- 4.
- 5.
Thomas Muirhead Flett (1923–1977). See [6092].
- 6.
Norman Levinson (1912–1975), professor at MIT.
- 7.
Zyoiti Suetuna (1898–1970), professor in Tokyo.
- 8.
Tikao Tatuzawa (1915–1997), professor in Nagoya and Tokyo.
- 9.
Hans Ludwig Hamburger (1889–1956), professor in Berlin, Ankara and Köln. See [2346].
- 10.
Guido Hoheisel (1894–1968), professor in Breslau and Köln.
- 11.
- 12.
Hans Rohrbach (1903–1993), professor in Göttingen and Mainz. See [5584].
- 13.
Hermann Zeitz (1870–1939), worked in a bank. See [678].
- 14.
Erik Johan Westzynthius (1901–1980), worked as actuary in an insurance company.
- 15.
Jacobus Hendricus van Lint (1932–2004), professor in Eindhoven. See [788].
- 16.
Albert Arnold Bennett (1888–1971), professor at the University of Texas, Lehigh University and Brown University.
- 17.
Heinz Hopf (1894–1971), professor at ETH Zürich, worked in algebraic topology. See [2812].
- 18.
Pierre Joseph Henry Baudet (1891–1921), professor at the Technical School at Delft. See [5843].
- 19.
- 20.
Leo Moser (1921–1970), professor at the University of Alberta. See [6764].
- 21.
Emma Lehmer (1906–2007). See [731].
- 22.
Vladimir Gennadievič Sprindžuk (1936–1987), professor in Minsk. See [3255].
- 23.
George Lomadze (1912–2005), professor in Tbilisi.
- 24.
- 25.
They used the bound O(n ε) for the number of representations of an integer n as the sum of two kth powers, proved by D. Cauer in [967].
- 26.
- 27.
The seventh paper of that series never appeared. It had to contain the proof that under a certain generalization of the Riemann Hypothesis one has lim inf (p n+1−p n )/log p n ≤2/3, p n being the nth consecutive prime.
- 28.
Erich Kamke (1890–1961), student of Landau. Professor in Tübingen (1926–1937 and 1946–1958). See [6540].
- 29.
- 30.
- 31.
Ralph Duncan James (1909–1979), professor at the University of Saskatchewan and University of British Columbia.
- 32.
Konstantin Konstantinovič Mardžanišvili (1903–1981), professor in Tbilisi. See [4617].
- 33.
Theodore Samuel Motzkin (1908–1970), professor in Jerusalem and at the University of California in Los Angeles.
- 34.
Otto Schreier (1901–1929), worked in Hamburg.
- 35.
Martin Eichler (1912–1992), professor in Münster, Marburg and Basel. See [3407].
- 36.
Paul Richard Halmos (1916–2006), professor at the University of Chicago, the University of Michigan, Indiana University and Santa Clara University. See [1946].
- 37.
For the Hasse principle see Sect. 3.4 below.
- 38.
Arnold Ephraim Ross (1906–2002), professor in St.Louis, at Notre Dame University and Ohio State University.
- 39.
Irving Kaplansky (1917–2006), professor in Chicago and Berkeley. See [347].
- 40.
A simpler proof was later given by E. Landau [3664].
- 41.
For a modern description of Weber’s ideas see G. Frei [2072].
- 42.
The narrow class-group of Z K consists of equivalence classes of ideals of Z K , two ideals I,J being equivalent if there exist totally positive elements α,β∈Z K , i.e., all of whose conjugates in real embeddings of k are positive, such that αI=βJ.
- 43.
Shokichi Iyanaga (1906–2006), professor in Tokyo.
- 44.
Wilhelm Magnus (1907–1990), professor in Göttingen, at the Courant Institute and Polytechnical Institute in New York. See [5].
- 45.
- 46.
A prime ideal ramifies in the extension L/K if it divides the relative discriminant of that extension.
- 47.
Erich Stiemke (1892–1915).
- 48.
Jacques Herbrand (1908–1931). See [1571].
- 49.
The restricted direct product of groups G n with respect to subgroups H n <G n is the subgroup of the direct product ∏ n G n consisting of elements (g n ) with almost all g n lying in H n .
- 50.
This book, along with the third volume of Landau’s book [3674], served as the main introduction to the theory of algebraic numbers for years to come.
- 51.
Martin Kneser (1928–2004), professor in Göttingen. See [5979].
- 52.
Čebotarev himself used this spelling of his name in papers published outside Russia.
- 53.
Arnold Scholz (1904–1942), worked in Freiburg and Kiel. See [6075].
- 54.
Yukata Taniyama (1927–1958), professor at the University of Tokyo. See [5704].
- 55.
Richard Dagobert Brauer (1901–1977), brother of Alfred Brauer, student of Schur, assistant in Königsberg, professor in Toronto, Ann Arbor and at Harvard University. See [5265].
- 56.
Bernard Dwork (1923–1998), professor at Johns Hopkins University, Princeton and Padua. See [3283].
- 57.
Albrecht Fröhlich (1916–2001), professor at King’s College, London. See [6077].
- 58.
He obtained his doctorate in 1920 in Marburg under supervision of K. Hensel.
- 59.
Hasse pointed out that essentially the same result can be deduced from a theorem by Minkowski [4320] which gives a complete set of invariants of quadratic forms under the action of invertible linear maps.
- 60.
Ernst Witt (1911–1991), professor in Hamburg. See [3305].
- 61.
Hans Reichardt (1908–1991), professor in Berlin. See [3429].
- 62.
Ernst Sejerstedt Selmer (1920–2006), professor in Bergen.
- 63.
Vasiliĭ Alekseevič Iskovskih (1939–2009), professor in Moscow. See [574].
- 64.
- 65.
William J. LeVeque (1923–2007), professor at the University of Michigan and Claremont Graduate University. See [4212].
- 66.
Felix Bernstein (1878–1956), professor in Göttingen, New York and Syracuse. See [2087].
- 67.
Rodion Osievič Kuzmin (1891–1949), professor in Perm and Leningrad. See [6385].
- 68.
Paul Pierre Lévy (1886–1971), professor in Paris. See [6091].
- 69.
Armand Borel (1923–2003), professor in Princeton. See [134].
- 70.
Ferdinand Lindemann (1852–1939), professor in Freiburg, Königsberg and Munich.
- 71.
Andreĭ Borisovič Šidlovskiĭ (1915–2007), student of Gelfond, professor in Moscow. See [3507].
- 72.
“I reserve a more precise presentation of the proofs sketched here for a later publication.”
- 73.
- 74.
Jan Popken (1905–1970), professor in Utrecht and Amsterdam. See [3094].
- 75.
- 76.
This lemma later found several different applications in number theory.
- 77.
Wilhelm Ljunggren (1905–1973), professor in Oslo and Bergen.
- 78.
Abraham Robinson (1918–1974), professor in Toronto, Jerusalem and at the University of California and Yale. See [4446].
- 79.
Dmitriĭ Konstantinovič Faddeev (1907–1989), professor in Leningrad. See [5367].
- 80.
“But the proof of the conjecture that every such equation, when its genus is larger than 1, has only finitely many solutions in rational numbers, will necessitate to overcome considerable difficulties” [5747, p. 34].
- 81.
Actually Poincaré described a geometrical procedure for generating new points of a curve from a finite number of points given, and defined the rank of the curve as the minimal number of its generators. He tacitly assumed this number to be finite, and on p. 173 formulated the question of which numbers are ranks of rational elliptic curves.
- 82.
The paper [6610], which is sometimes quoted as the source of the Mordell–Weil theorem, contains only a fresh proof of Mordell’s result for the base field Q.
- 83.
See also Delone, B.N.
- 84.
See also Delaunay, B.
- 85.
The same pagination occurring in two places is not a result of a printing error, but occurred really.
- 86.
The same pagination occurring in two places is not a result of a printing error, but occurred really.
References
Abikoff, W., Birman, J.S., Kuiken, K. (eds.): The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions. Contemp. Math., vol. 169. Am. Math. Soc., Providence (1994)
Agapito, J.: Weighted Brianchon-Gram decomposition. Can. Math. Bull. 49, 161–169 (2006)
Aleksentsev, Yu.M.: On the measure of approximation of π by algebraic numbers. Mat. Zametki 66, 483–493 (1999) (in Russian)
Alzer, H.: On rational approximation to e. J. Number Theory 68, 57–62 (1998)
Andrews, G.E.: An asymptotic expression for the number of solutions of a general class of Diophantine equations. Trans. Am. Math. Soc. 99, 272–277 (1961)
Andrews, G.E.: A lower bound for the volume of strictly convex bodies with many boundary lattice points. Trans. Am. Math. Soc. 106, 270–279 (1963)
Anglin, W.S.: Simultaneous Pell equations. Math. Comput. 65, 355–359 (1996)
Aramata, H.: Über die Teilbarkeit der Zetafunktionen gewisser algebraischer Zahlkörper. Proc. Imp. Acad. (Tokyo) 7, 334–336 (1931)
Aramata, H.: Über die Teilbarkeit der Dedekindschen Zetafunktionen. Proc. Imp. Acad. (Tokyo) 9, 31–34 (1933)
Arkhipov, G.I.: The Hilbert-Kamke problem. Izv. Akad. Nauk SSSR, Ser. Mat. 48, 3–52 (1984) (in Russian)
Arkhipov, G.I., Karatsuba, A.A., Čubarikov, V.N.: Multiple trigonometric sums. Tr. Mat. Inst. Steklova 151, 1–128 (1980) (in Russian)
Arthur, J., et al.: Armand Borel (1923–2003). Not. Am. Math. Soc. 51, 498–524 (2004)
Artin, E.: Über die Zetafunktionen gewisser algebraischer Zahlkörper. Math. Ann. 89, 147–156 (1923) [[144], pp. 95–104]
Artin, E.: Über eine neue Art von L-Reihen. Abh. Math. Semin. Univ. Hamb. 3, 89–108 (1924) [[144], pp. 105–124]
Artin, E.: Beweis des allgemeinen Reziprozitätsgesetzes. Abh. Math. Semin. Univ. Hamb. 5, 353–363 (1927) [[144], pp. 131–141]
Artin, E.: Über die Zerlegung definiter Funktionen in Quadrate. Abh. Math. Semin. Univ. Hamb. 5, 100–115 (1927) [[144], pp. 273–288]
Artin, E.: Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz. Abh. Math. Semin. Univ. Hamb. 7, 46–51 (1930) [[144], pp. 159–164]
Artin, E.: Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren. Abh. Math. Semin. Univ. Hamb. 8, 292–306 (1930) [[144], pp. 165–179]
Artin, E., Schreier, O.: Algebraische Konstruktion reeller Körper. Abh. Math. Semin. Univ. Hamb. 5, 85–99 (1927) [[144], pp. 258–272]
Artin, E., Tate, J.: Class Field Theory. Benjamin, Elmsford (1968); 2nd ed. Addison-Wesley, 1990 [Reprint: Chelsea (2009)]
Arwin, A.: Common solutions of two simultaneous Pell equations. Ann. Math. 23, 307–312 (1923)
Athanasiadis, C.A.: Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley. J. Reine Angew. Math. 583, 163–174 (2005)
Atkinson, F.V.: The mean-value of the Riemann zeta-function. Acta Math. 81, 353–376 (1949)
Ayoub, R.[G.]: On Rademacher’s extension of the Goldbach-Vinogradoff theorem. Trans. Am. Math. Soc. 74, 482–491 (1953)
Bach, E., Sorenson, J.: Explicit bounds for primes in residue classes. Math. Comput. 65, 1717–1735 (1996)
Backlund, R.J.: Über die Differenzen zwischen den Zahlen, die zu den n ersten Primzahlen teilerfremd sind. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 32(2), 1–9 (1929)
Baer, W.S.: Über die Zerlegung der ganzen Zahlen in sieben Kuben. Math. Ann. 74, 511–514 (1913)
Baker, A., Schmidt, W.M.: Diophantine approximation and Hausdorff dimension. Proc. Lond. Math. Soc. 21, 1–11 (1970)
Baker, R.C.: Sprindzuk’s theorem and Hausdorff dimension. Mathematika 23, 184–197 (1976)
Baker, R.C.: Fractional parts of several polynomials. Q. J. Math. 28, 453–471 (1977)
Baker, R.C.: Fractional parts of several polynomials, II. Mathematika 25, 76–93 (1978)
Baker, R.C.: On the fractional parts of αn 2 and βn. Glasg. Math. J. 22, 181–183 (1981)
Baker, R.C.: Weyl sums and Diophantine approximation. J. Lond. Math. Soc. 25, 25–34 (1982)
Baker, R.C.: On the fractional parts of αn 3, βn 2 and γn. In: Journées Arithmétiques 1980, Exeter, 1980, pp. 226–231. Cambridge University Press, Cambridge (1982)
Baker, R.C.: Diophantine Inequalities. Clarendon, Oxford (1986)
Baker, R.C., Brüdern, J., Harman, G.: The fractional part of αn k for square-free n. Q. J. Math. 42, 421–431 (1991)
Baker, R.C., Brüdern, J., Harman, G.: Simultaneous Diophantine approximation with square-free numbers. Acta Arith. 63, 51–60 (1993)
Baker, R.C., Gajraj, J.: Some non-linear Diophantine approximations. Acta Arith. 31, 325–341 (1976)
Baker, R.C., Harman, G.: The difference between consecutive primes. Proc. Lond. Math. Soc. 72, 261–280 (1996)
Baker, R.C., Harman, G., Pintz, J.: The difference between consecutive primes, II. Proc. Lond. Math. Soc. 83, 532–562 (2001)
Baker, R.C., Schlickewei, H.P.: Indefinite quadratic forms. Proc. Lond. Math. Soc. 54, 385–411 (1987)
Balasubramanian, R.: An improvement on a theorem of Titchmarsh on the mean square of \(|\zeta\frac{1}{2}+it)|\). Proc. Lond. Math. Soc. 36, 540–576 (1978)
Balasubramanian, R., Ramachandra, K.: On the frequency of Titchmarsh’s phenomenon for ζ(s). III. Proc. Indian Acad. Sci. Math. Sci. 86, 341–351 (1977)
Balasubramanian, R., Ramachandra, K.: Proof of some conjectures on the mean-value of Titchmarsh series, I, Hardy. Ramanujan J. 13, 1–20 (1990)
Balog, A., Perelli, A.: Diophantine approximation by square-free numbers. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 11, 353–359 (1984)
Bartz, K.: An effective order of Hecke-Landau zeta functions near the line σ=1, I. Acta Arith. 50, 183–193 (1988); corr. 58 211 (1991)
Bartz, K.: An effective order of Hecke-Landau zeta functions near the line σ=1, II. (Some applications). Acta Arith. 52, 163–170 (1989)
Bartz, K.: On an effective order estimate of the Riemann zeta function in the critical strip. Monatshefte Math. 109, 267–270 (1990)
Barvinok, A.I.: Computing the Ehrhart polynomial of a convex lattice polytope. Discrete Comput. Geom. 12, 35–48 (1994)
Bass, H., Lam, T.Y.: Irving Kaplansky 1917–2006. Not. Am. Math. Soc. 54, 1477–1499 (2007)
Bateman, P.T.: On the representation of a number as the sum of three squares. Trans. Am. Math. Soc. 71, 70–101 (1951)
Baxa, C.: On the growth of the denominators of convergents. Acta Math. Acad. Sci. Hung. 83, 125–130 (1999)
Baxa, C.: Lévy constants of transcendental numbers. Proc. Am. Math. Soc. 137, 2243–2249 (2009)
Belhoste, B.: Cauchy, 1789–1857. Paris (1985). [English translation: Augustin-Louis Cauchy. A biography. Springer, 1991]
Bennett, A.A.: On sets of three consecutive integers which are quadratic residues of primes. Bull. Am. Math. Soc. 31, 411–412 (1925)
Bennett, A.A.: Large primes have four consecutive quadratic residues. Tohoku Math. J. 27, 53–57 (1926)
Bennett, A.A.: Large primes have at least five consecutive quadratic residues. Bull. Am. Math. Soc. 32, 33 (1926)
Bennett, M.A.: Solving families of simultaneous Pell equations. J. Number Theory 67, 246–251 (1997)
Bennett, M.A.: On the number of solutions of simultaneous Pell equations. J. Reine Angew. Math. 498, 173–199 (1998)
Bennett, M.A.: Rational approximation to algebraic number of small height; the Diophantine equation |ax n−by n|=1. J. Reine Angew. Math. 535, 1–49 (2001)
Bennett, M.A.: On the representation of unity by binary cubic forms. Trans. Am. Math. Soc. 353, 1507–1534 (2001)
Bennett, M.A., Cipu, M., Mignotte, M., Okazaki, R.: On the number of solutions of simultaneous Pell equations, II. Acta Arith. 122, 407–417 (2006)
Bennett, M.A., de Weger, B.M.M.: On the Diophantine equation |ax n−by n|=1. Math. Comput. 67, 413–438 (1998)
Bentkus, V., Götze, F.: Lattice point problems and distribution of values of quadratic forms. Ann. Math. 150, 977–1027 (1999)
Bergman, G.: On the exceptional group of a Weierstrass curve in an algebraic number field. Acta Math. 91, 113–124 (1954)
Berlekamp, E.R.: A construction for partitions which avoid long arithmetic progressions. Can. Math. Bull. 11, 409–414 (1968)
Bernik, V.I.: The Baker-Schmidt conjecture. Dokl. Akad. Nauk Belorus. 23, 392–395 (1979) (in Russian)
Bernik, V.I.: Application of the Hausdorff dimension in the theory of Diophantine approximations. Acta Arith. 42, 219–253 (1983) (in Russian)
Bernik, V.I., Melničuk, Yu.V.: Diophantine approximations and Hausdorff dimension. Nauka i Tekhnika, Minsk (1988) (in Russian)
Bernstein, F.: Über eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem. Math. Ann. 71, 417–439 (1912)
Besicovitch, A.S.: Sets of fractional dimension. IV. On rational approximation to real numbers. J. Lond. Math. Soc. 9, 126–131 (1934)
Beukers, F.: The lattice-points of n-dimensional tetrahedra. Indag. Math. 37, 365–372 (1975)
Beukers, F., Bézivin, J.-P., Robba, P.: An alternative proof of the Lindemann-Weierstrass theorem. Am. Math. Mon. 97, 193–197 (1990)
Beukers, F., Brownawell, W.D., Heckman, G.: Siegel normality. Ann. Math. 127, 279–308 (1988)
Bhargava, M.: On the Conway-Schneeberger Fifteen Theorem. In: Contemp. Math., vol. 272, pp. 27–37. Am. Math. Soc., Providence (2000)
Bhargava, M.: Finiteness theorem for quadratic forms, to appear
Bhargava, M., Hanke, J.: Universal quadratic forms and the 290-theorem. Invent. Math., to appear
Bierstedt, R.G., Mills, W.H.: On the bound for a pair of consecutive quartic residues of a prime. Proc. Am. Math. Soc. 14, 628–632 (1963)
Billing, G.: Beiträge zur arithmetischen Theorie der ebenen kubischen Kurven vom Geschlechte Eins. Nova Acta R. Soc. Sci. Ups. 11(1), 1–165 (1938)
Bilu, Y.[F.], Bugeaud, Y.: Démonstration du théorème de Baker-Feldman via les formes linéaires en deux logarithmes. J. Théor. Nr. Bordx. 12, 13–23 (2000)
Birch, B.J.: Forms in many variables. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 265, 245–263 (1961/1962)
Birch, B.J., Davenport, H.: Indefinite quadratic forms in many variables. Mathematika 5, 8–12 (1958) [[1380], vol. 3, pp. 1077–1081]
Birch, B.J., Swinnerton-Dyer, H.P.F.: The Hasse problem for rational surfaces. J. Reine Angew. Math. 274/275, 164–174 (1975)
Bochnak, J., Oh, B.-K.: Almost regular quaternary quadratic forms. Ann. Inst. Fourier 58, 1499–1549 (2008)
Bochnak, J., Oh, B.-K.: Almost-universal quadratic forms: an effective solution of a problem of Ramanujan. Duke Math. J. 147, 131–156 (2009)
Bogomolov, F., et al.: Vassiliĭ Alekseevič Iskovskih. Usp. Mat. Nauk 64(5), 167–174 (2009) (in Russian)
Bohr, H., Landau, E.: Über das Verhalten von ζ(s) und ζ κ (s) in der Nähe der Geraden σ=1. Nachr. Ges. Wiss. Göttingen, 1910, 303–330. [[578], vol. 3, B1; [3680], vol. 4, pp. 221–248]
Boklan, K.D.: The asymptotic formula in Waring’s problem. Mathematika 41, 329–347 (1994)
Bombieri, E.: On a theorem of van Lint and Richert. In: Symposia Mathematica, vol. 4, pp. 175–180. Academic Press, London (1970)
Bombieri, E.: Effective Diophantine approximation on G m . Ann. Sc. Norm. Super. Pisa, Cl. Sci. 20, 61–89 (1993)
Bombieri, E., Iwaniec, H.: On the order of \(\zeta(\frac{1}{2}+it)\). Ann. Sc. Norm. Super. Pisa, Cl. Sci. 13, 449–472 (1986)
Bombieri, E., Iwaniec, H.: Some mean-value theorems for exponential sums. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 13, 473–486 (1986)
Bombieri, E., Pila, J.: The number of integral points on arcs and ovals. Duke Math. J. 59, 337–357 (1989)
Bombieri, E., Vaaler, J.: On Siegel’s lemma. Invent. Math. 73, 11–32 (1983); add.: 75, 377 (1984)
Borel, A.: Values of indefinite quadratic forms at integral points and flows on spaces of lattices. Bull. Am. Math. Soc. 32, 184–204 (1995)
Borel, A., Chowla, S., Herz, C.S., Iwasawa, K., Serre, J.-P.: Seminar on Complex Multiplication. Lecture Notes in Math., vol. 21. Springer, Berlin (1966)
Borel, É.: Sur la nature arithmétique du nombre e. C. R. Acad. Sci. Paris 128, 596–599 (1899)
Borel, É.: Les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Mat. Palermo 27, 247–271 (1909)
Borel, É.: Sur un problème de probabilités relatif aux fractions continues. Math. Ann. 72, 578–584 (1912)
Borevič, Z.I.: On the proof of the principal ideal theorem. Vestn. Leningr. Univ. 12(13), 5–8 (1957) (in Russian)
Boutin, A., Teilhet, P.F.: L’Intermédiaire des Math., 11, 68, 182 (1904)
Brauer, A.: Über einige spezielle diophantische Gleichungen. Math. Z. 25, 499–504 (1926)
Brauer, A.: Über Sequenzen von Potenzresten. SBer. Preuß. Akad. Wiss. Berlin, 1928, 9–16
Brauer, A.: Über diophantische Gleichungen mit endlich vielen Lösungen. J. Reine Angew. Math. 160, 70–99 (1929)
Brauer, A.: Über die Verteilung der Potenzreste. Math. Z. 35, 39–50 (1932)
Brauer, A.: Nachruf auf Hermann Zeitz. SBer. Berliner Math. Ges. 33, 2–6 (1934)
Brauer, A., Zeitz, H.: Über eine zahlentheoretische Behauptung von Legendre. SBer. Berliner Math. Ges. 29, 116–125 (1930)
Brauer, R.: On the zeta-functions of algebraic number fields. Am. J. Math. 69, 243–250 (1947)
Brauer, R.: On the zeta-functions of algebraic number fields, II. Am. J. Math. 72, 739–746 (1950)
Brauer, R.: On Artin’s L-series with general group characters. Ann. Math. 48, 502–514 (1947)
Brent, R.P.: The first occurrence of large gaps between successive primes. Math. Comput. 27, 959–963 (1973)
Brent, R.P.: On the zeros of the Riemann zeta-function in the critical strip. Math. Comput. 33, 1361–1372 (1979)
Brent, R.P.: The first occurrence of certain large prime gaps. Math. Comput. 35, 1435–1436 (1980)
Brent, R.P., van de Lune, J., te Riele, H.J.J., Winter, D.T.: On the zeros of the Riemann zeta-function in the critical strip, II. Math. Comput. 39, 681–688 (1982); corr.: 46, 771 (1986)
Breusch, R.: Zur Verallgemeinerung des Bertrandschen Postulates, daß zwischen x und 2x stets Primzahlen liegen. Math. Z. 34, 505–526 (1932)
Brillhart, J.: Emma Lehmer (1906–2007). Not. Am. Math. Soc. 54, 1500–1501 (2007)
Brillhart, J., Lehmer, D.H., Lehmer, E.: Bounds for pairs of consecutive seventh and higher power residues. Math. Comput. 18, 397–407 (1964)
Brion, M.: Points entiers dans les polyèdres convexes. Ann. Sci. Éc. Norm. Super. 21, 653–663 (1988)
Brion, M.: Polyèdres et réseaux. Enseign. Math. 38, 71–88 (1992)
Brion, M.: Points entiers dans les polytopes convexes. Astérisque 227, 145–169 (1995)
Brion, M., Vergne, M.: Residue formulae, vector partition functions and lattice points in rational polytopes. J. Am. Math. Soc. 10, 797–833 (1997)
Browning, T.D., Heath-Brown, D.R.: Rational points on quartic hypersurfaces. J. Reine Angew. Math. 629, 37–88 (2009)
de Bruijn, N.G.: The asymptotic behaviour of a function occurring in the theory of primes. J. Indian Math. Soc. (N.S.) 15, 25–32 (1951)
de Bruijn, N.G.: On the number of positive integers ≤x and free of prime factors >y. Indag. Math. 15, 50–60 (1951)
de Bruijn, N.G.: On the number of positive integers ≤x and free of prime factors >y, II. Indag. Math. 28, 239–247 (1966)
de Bruijn, N.G.: In memoriam Jack van Lint (1932–2004). Nieuw Arch. Wiskd. 6, 105–109 (2005)
Buhler, J.P.: Icosahedral Galois Representations. Lecture Notes in Math., vol. 654. Springer, Berlin (1978)
Buhštab, A.A.: Asymptotical evaluation of a general number-theoretical function. Mat. Sb. 2, 1239–1246 (1937) (in Russian)
Buhštab, A.A.: On those numbers in an arithmetic progression all prime factors of which are small in order of magnitude. Dokl. Akad. Nauk SSSR 67, 5–8 (1949) (in Russian)
Buhštab, A.A.: On an asymptotic estimate of the number of numbers of an arithmetic progression which are not divisible by “relatively” small prime numbers. Mat. Sb. 28, 165–184 (1951) (in Russian)
Bundschuh, P.: Irrationalitätsmaße für e a, a≠0 rational oder Liouville-Zahl. Math. Ann. 192, 229–242 (1971)
Bundschuh, P.: Einführung in die Zahlentheorie. Springer, Berlin (1988)
Burckhardt, J.J.: Rudolf Fueter. Vierteljahr. Naturforsch. Ges. Zürich 95, 284–287 (1950)
Bureau, J., Morales, J.: A note on indefinite ternary quadratic forms representing all odd integers. Bol. Soc. Parana. Mat. 23, 85–92 (2005)
Burgess, D.A.: The distribution of quadratic residues and non-residues. Mathematika 4, 106–112 (1957)
Burgess, D.A.: On character sums and primitive roots. Proc. Lond. Math. Soc. 12, 179–192 (1962)
Burgess, D.A.: A note on the distribution of residues and non-residues. J. Lond. Math. Soc. 38, 253–256 (1963)
Buzzard, K., Dickinson, M., Shepherd-Barron, N., Taylor, R.: On icosahedral Artin representations. Duke Math. J. 109, 283–318 (2001)
Buzzard, K., Stein, W.A.: A mod five approach to modularity of icosahedral Galois representations. Pac. J. Math. 203, 265–282 (2002)
Buzzard, K., Taylor, R.: Companion forms and weight one forms. Ann. Math. 149, 905–919 (1999)
Canfield, E.R., Erdős, P., Pomerance, C.: On a problem of Oppenheim concerning “factorisatio numerorum”. J. Number Theory 17, 1–28 (1983)
Cantor, D.C., Roquette, P.: On Diophantine equations over the ring of all algebraic integers. J. Number Theory 18, 1–26 (1984)
Cappell, S.E., Shaneson, J.L.: Genera of algebraic varieties and counting of lattice points. Bull. Am. Math. Soc. 30, 62–69 (1994)
Carlson, F.: Über die Nullstellen der Dirichletschen Reihen und der Riemannscher ζ-Funktion. Ark. Mat. Astron. Fys. 15(20), 1–28 (1920)
Carlson, F.: Contributions á la théorie des séries de Dirichlet, I. Ark. Mat. Astron. Fys. 16(18), 1–19 (1922)
Carlson, F.: Contributions á la théorie des séries de Dirichlet, II. Ark. Mat. Astron. Fys. 19(25), 1–17 (1926)
Cassels, J.W.S.: The rational solutions of the diophantine equation Y 2=X 3−D. Acta Math. 82, 243–273 (1950); corr. 84, 299 (1951)
Cassels, J.W.S.: An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1957) [Reprint: Hafner, 1972]
Cassels, J.W.S.: Arithmetic on curves of genus 1, I. On a conjecture of Selmer. J. Reine Angew. Math. 202, 52–99 (1959)
Cassels, J.W.S.: Mordell’s finite basis theorem revisited. Math. Proc. Camb. Philos. Soc. 100, 31–41 (1986)
Cassels, J.W.S., Guy, M.J.T.: On the Hasse principle for cubic surfaces. Mathematika 13, 111–120 (1966)
Cassou-Noguès, Ph., Chinburg, T., Fröhlich, A., Taylor, M.J.: L-functions and Galois modules. In: L-functions and Arithmetic, Durham, 1989, pp. 75–139. Cambridge University Press, Cambridge (1991)
Catlin, P.A.: Two problems in metric Diophantine approximation, I. J. Number Theory 8, 282–288 (1976)
Catlin, P.A.: Two problems in metric Diophantine approximation, II. J. Number Theory 8, 289–297 (1976)
Cauchy, A.: Démonstration du théorème genéral de Fermat sur les nombres polygones. Mem. Sci. Math. Phys. Inst. Fr. 14, 177–220 (1813–1815) [Oeuvres, (2), vol. 6, pp. 320–353, Paris, 1887]
Cauer, D.: Neue Anwendungen der Pfeifferschen Methode zur Abschätzung zahlentheoretischer Funktionen. Dissertation, Göttingen (1914)
Čebyšev, P.L.: Mémoire sur nombres premiers. Mémoires des savants étrangers de l’Acad. Sci. St. Pétersbourg 7, 17–33 (1850) [J. math. pures appl., 17, 366–390 (1852); Oeuvres, vol. 1, pp. 49–70, S. Pétersbourg 1899, reprint: Chelsea, 1962; [973], vol. 1, pp. 191–207 (in Russian)]
Chan, W.K., Oh, B.-K.: Positive ternary quadratic forms with finitely many exceptions. Proc. Am. Math. Soc. 132, 1567–1573 (2004)
Chandrasekharan, K., Narasimhan, R.: The approximate functional equation for a class of zeta-functions. Math. Ann. 152, 30–64 (1963)
Chang, K.L.: On some Diophantine equations y 2=x 3+k with no rational solutions. Q. J. Math. 19, 181–188 (1948)
Chen, B.: Lattice points, Dedekind sums, and Ehrhart polynomials of lattice polyhedra. Discrete Comput. Geom. 28, 175–199 (2002)
Cheng, Y.F.: Explicit estimate on primes between consecutive cubes. Rocky Mt. J. Math. 40, 117–153 (2010)
Cheng, Y.F., Graham, S.W.: Explicit estimates for the Riemann zeta-function. Rocky Mt. J. Math. 34, 1261–1289 (2004)
Chevalley, C.: La théorie du symbole de restes normiques. J. Reine Angew. Math. 169, 140–157 (1933)
Chevalley, C.: Sur la théorie du corps de classes dans les corps finis et les corps locaux. J. Fac. Sci. Univ. Tokyo 2, 365–476 (1933)
Chevalley, C.: Généralisation de la théorie du corps de classes pour les extensions infinies. J. Math. Pures Appl. 15, 359–371 (1936)
Chevalley, C.: La théorie du corps de classes. Ann. Math. 41, 394–418 (1940)
Chevalley, C., Nehrkorn, H.: Sur les démonstrations arithmétiques dans la théorie du corps de classes. Math. Ann. 111, 364–371 (1935)
Chowla, I.: A new evaluation of the number Γ(k) in Waring’s problem. Proc. Indian Acad. Sci. Math. Sci. 6, 97–103 (1937)
Chowla, S.D.: On the greatest prime factor of a certain product. J. Indian Math. Soc. 18, 135–137 (1929)
Chowla, S.: A theorem on irrational indefinite quadratic forms. J. Lond. Math. Soc. 9, 162–163 (1934)
Chowla, S.: The representation of a number as a sum of four squares and a prime. Acta Arith. 1, 115–122 (1935)
Chowla, S.: Review of [2497]. Math. Rev. 28, #1179
Chowla, S., Pillai, S.S.: Hypothesis K of Hardy and Littlewood. Math. Z. 41, 537–540 (1936)
Chowla, S., Vijayaraghavan, T.: On the largest prime divisors of numbers. J. Indian Math. Soc. 11, 31–37 (1947)
Cigler, J.: Über eine Verallgemeinerung des Hauptsatzes der Theorie der Gleichverteilung. J. Reine Angew. Math. 210, 141–147 (1962)
Cijsouw, P.L.: A transcendence measure for π. In: Transcendence Theory: Advances and Applications, pp. 93–100. Academic Press, San Diego (1977)
Cipu, M., Mignotte, M.: On the number of solutions to simultaneous hyperbolic Diophantine equations. J. Number Theory 125, 365–392 (2007)
Coates, J.: An effective p-adic analogue of a theorem of Thue, II. The greatest prime factor of a binary form. Acta Arith. 16, 399–412 (1969/1970)
Cohen, H.: Number Theory, vols. I–II. Springer, Berlin (2007)
Coleman, M.D.: On the equation b 1 p−b 2 P 2=b 3. J. Reine Angew. Math. 403, 1–66 (1990)
Colliot-Thélène, J.-L.: L’arithmétique des variétés rationnelles. Ann. Fac. Sci. Toulouse 1, 295–336 (1992)
Colliot-Thélène, J.-L., Kanevsky, D., Sansuc, J.-J.: Arithmétique des surfaces cubiques diagonales. In: Lecture Notes in Math., vol. 1290, pp. 1–108. Springer, Berlin (1987)
Colliot-Thélène, J.-L., Poonen, B.: Algebraic families of nonzero elements of Shafarevich-Tate groups. J. Am. Math. Soc. 13, 83–99 (2000)
Colliot-Thélène, J.-L., Sansuc, J.-J.: Sur le principle de Hasse et l’approximation faible, et sur une hypothèse de Schinzel. Acta Arith. 41, 34–53 (1982)
Colliot-Thélène, J.-L., Swinnerton-Dyer, P.: Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties. J. Reine Angew. Math. 453, 49–112 (1994)
Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M., Snaith, N.C.: Integral moments of L-functions. Proc. Lond. Math. Soc. 91, 33–104 (2005)
Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Lower order terms in the full moment conjecture for the Riemann zeta function. J. Number Theory 128, 1516–1554 (2008)
Conrey, J.B., Ghosh, A.: On mean values of the zeta-function. Mathematika 31, 159–161 (1984)
Conrey, J.B., Ghosh, A.: On mean values of the zeta-function, II. Acta Arith. 52, 367–371 (1989)
Conrey, J.B., Ghosh, A.: On mean values of the zeta-function, III. In: Proceedings of the Conference on Analytic Number Theory, Maiori, 1989, pp. 35–59. Universit’a di Salerno, Salerno (1992)
Conrey, J.B., Gonek, S.M.: High moments of the Riemann zeta-function. Duke Math. J. 107, 577–604 (2001)
Conway, J.H.: Universal quadratic forms and the fifteen theorem. In: Contemp. Math., vol. 272, pp. 23–26. Am. Math. Soc., Providence (2000)
Cook, R.J.: On the fractional parts of a set of points. Mathematika 19, 63–68 (1972)
Cook, R.J.: On the fractional parts of a set of points, II. Pac. J. Math. 45, 81–85 (1973)
Cook, R.J.: On the fractional parts of a set of points, III. J. Lond. Math. Soc. 9, 490–494 (1974/1975)
Cook, R.J.: On the fractional parts of a set of points, IV. Indian J. Math. 19, 7–23 (1977)
Cook, R.J.: On the fractional parts of a polynomial. Can. J. Math. 28, 168–173 (1976)
Cook, R.J.: Small fractional parts of quadratic forms in many variables. Mathematika 27, 25–29 (1980)
Corvaja, P., Zannier, U.: A subspace theorem approach to integral points on curves. C. R. Acad. Sci. Paris 334, 267–271 (2002)
Cougnard, J.: Les travaux de A. Fröhlich, Ph. Cassou-Noguès et M.J. Taylor sur les bases normales. Astérisque 105/106, 25–38 (1983)
Cramér, H.: Some theorems concerning prime numbers. Ark. Math. Astron. Fys. 15(5), 1–33 (1921) [[1274], vol. 1, pp. 138–170]
Cramér, H.: On the distribution of primes. Proc. Camb. Philos. Soc. 20, 272–280 (1921) [[1274], vol. 1, pp. 171–179]
Croot, E.S. III: On a coloring conjecture about unit fractions. Ann. Math. 157, 545–556 (2003)
Čubarikov, V.N.: Simultaneous representation of natural numbers by sums of powers of primes. Dokl. Akad. Nauk SSSR 286, 828–831 (1986) (in Russian)
Cusick, T.W.: Continuants with bounded digits, I. Mathematika 24, 166–172 (1977)
Cusick, T.W.: Continuants with bounded digits, II. Mathematika 25, 107–109 (1978)
Cusick, T.W.: Continuants with bounded digits, III. Monatshefte Math. 99, 105–109 (1985)
Cutter, P.A.: Finding prime pairs with particular gaps. Math. Comput. 70, 1737–1744 (2001)
Dani, S.G.: A proof of Margulis’ theorem on values of quadratic forms, independent of the axiom of choice. Enseign. Math. 40, 49–58 (1994)
Dani, S.G., Margulis, G.A.: Values of quadratic forms at primitive integral points. Invent. Math. 98, 405–424 (1989)
Dani, S.G., Margulis, G.A.: Values of quadratic forms at integral points: An elementary approach. Enseign. Math. 36, 143–174 (1990)
Danicic, I.: An extension of a theorem of Heilbronn. Mathematika 5, 30–37 (1958)
Danicic, I.: On the fractional parts of θx 2 and φx 2. J. Lond. Math. Soc. 34, 353–357 (1959)
Danset, R.: Méthode du cercle adélique et principe de Hasse fin pour certains systémes de formes. Enseign. Math. 31, 1–66 (1985)
Dartyge, C.: Entiers de la forme n 2+1 sans grand facteur premier. Acta Math. Acad. Sci. Hung. 72, 1–34 (1996)
Dartyge, C., Martin, G., Tenenbaum, G.: Polynomial values free of large prime factors. Period. Math. Hung. 43, 111–119 (2001)
Davenport, H.: On the distribution of l-th power residues (mod p). J. Lond. Math. Soc. 7, 117–121 (1932) [[1380], vol. 4, pp. 1457–1461]
Davenport, H.: On Waring’s problem for fourth powers. Ann. Math. 40, 731–747 (1939) [[1380], vol. 3, pp. 946–962]
Davenport, H.: On Waring’s problem for cubes. Acta Math. 71, 123–143 (1939) [[1380], vol. 3, pp. 925–945]
Davenport, H.: Indefinite quadratic forms in many variables. Mathematika 3, 81–101 (1956) [[1380], vol. 3, pp. 1035–1055]
Davenport, H.: Indefinite quadratic forms in many variables, II. Proc. Lond. Math. Soc. 8, 109–126 (1958) [[1380], vol. 3, pp. 1058–1075]
Davenport, H.: Analytic Methods for Diophantine Equations and Diophantine Inequalities. University of Michigan Press, Ann Arbor (1962); 2nd ed. Cambridge, 2005
Davenport, H.: On a theorem of Heilbronn. Q. J. Math. 18, 339–344 (1967) [[1380], vol. 3, pp. 1307–1312]
Davenport, H., Erdős, P.: The distribution of quadratic and higher residues. Publ. Math. (Debr.) 2, 252–265 (1952) [[1380], vol. 4, pp. 1562–1575]
Davenport, H., Heilbronn, H.: On Waring’s problem for fourth powers. Proc. Lond. Math. Soc. 41, 143–150 (1936) [[1380], vol. 3, pp. 875–882; [2715], pp. 283–290]
Davenport, H., Heilbronn, H.: On indefinite quadratic forms in five variables. J. Lond. Math. Soc. 21, 185–193 (1946) [[1380], vol. 3, pp. 1010–1018; [2715], pp. 360–368]
Davenport, H., Ridout, D.: Indefinite quadratic forms. Proc. Lond. Math. Soc. 9, 544–555 (1959) [[1380], vol. 3, pp. 1105–1116]
Dedekind, R.: Schreiben an Herrn Borchardt über die Theorie der elliptischen Modulfunktionen. J. Reine Angew. Math. 83, 265–292 (1877) [[1423], vol. 1, pp. 174–201]
Dedekind, R.: Über die Anzahl von Idealklassen in reinen kubischen Zahlkörpern. J. Reine Angew. Math. 121, 40–123 (1900) [[1423], vol. 2, pp. 148–233]
de la Bretèche, R., Tenenbaum, G.: Entiers friables: inégalité de Turán-Kubilius et applications. Invent. Math. 159, 531–588 (2005)
de la Bretèche, R., Tenenbaum, G.: Propriétés statistiques des entiers friables. Ramanujan J. 9, 139–202 (2005)
Delaunay, B.Footnote
See also Delone, B.N.
: La solution générale de l’équation X 3 ϱ+Y 3=1. C. R. Acad. Sci. Paris 162, 150–151 (1916)Delaunay, B.: Sur la représentation des nombres par les formes binaires. C. R. Acad. Sci. Paris 178, 1458–1461 (1924)
Delaunay, B.: Vollständige Lösung der unbestimmten Gleichung X 3 q+Y 3=1 in ganzen Zahlen. Math. Z. 28, 1–9 (1928)
Delaunay, B.: Über die Darstellung der Zahlen durch die binären kubischen Formen von negativer Diskriminante. Math. Z. 31, 1–26 (1930).
Deligne, P.: Les constantes des équations fonctionnelles des fonctions L. In: Lecture Notes in Math., vol. 349, pp. 501–597. Springer, Berlin (1973)
Delone, B.N.Footnote
See also Delaunay, B.
: Solving the indeterminate equation X 3 ϱ+Y 3=1. Comm. Soc. Math. Charkov, (2) 15, 1915/1916, 1–16, 46–48, 75–76 (in Russian)Delone, B.N., Faddeev, D.K.: The theory of irrationalities of third degree. Tr. Mat. Inst. Steklova 11, 1–340 (1940) (in Russian). [English translation, Am. Math. Soc., 1964]
Demyanenko, V.A.: Rational points of a class of algebraic curves. Izv. Akad. Nauk SSSR, Ser. Mat. 30, 1373–1396 (1966) (in Russian)
Demyanenko, V.A.: Rational points of certain curves of higher genus. Acta Arith. 12, 333–354 (1966/1967)
Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 (1982/1983)
Deshouillers, J.-M., Iwaniec, H.: On the greatest prime factor of n 2+1. Ann. Inst. Fourier 32(4), 1–11 (1982)
Deuring, M.: Zur arithmetischen Theorie der algebraischen Funktionen. Math. Ann. 106, 77–102 (1932)
Deuring, M.: Über den Tschebotareffschen Dichtigkeitssatz. Math. Ann. 110, 414–415 (1935)
Deuring, M.: Algebraische Begründung der komplexen Multiplikation. Abh. Math. Semin. Univ. Hamb. 16, 32–47 (1949)
Deuring, M.: Die Struktur der elliptischen Funktionenkörper und die Klassenkörper der imaginären quadratischer Zahlkörper. Math. Ann. 124, 393–426 (1952)
Deuring, M.: Die Klassenkörper der komplexen Multiplikation. In: Enzyklopädie der Mathematischen Wissenschaften, vol. I2, Heft 10. Teubner, Leipzig (1958)
Diaconu, A., Goldfeld, D., Hoffstein, J.: Multiple Dirichlet series and moments of zeta and L-functions. Compos. Math. 139, 197–360 (2003)
Diaz, R., Robins, S.: The Ehrhart polynomial of a lattice polytope. Ann. Math. 145, 503–518 (1997); err.: vol. 146, 1997, p. 237
Dickmann, K.: On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. 22(10), 1–14 (1930)
Dickson, L.E.: History of the Theory of Numbers. Carnegie Institution of Washington, Washington (1919) [Reprints: Chelsea, 1952, 1966]
Dickson, L.E.: Quaternary quadratic forms representing all integers. Am. J. Math. 49, 39–56 (1927)
Dickson, L.E.: All positive integers are sums of values of a quadratic function of x. Bull. Am. Math. Soc. 33, 713–720 (1927)
Dickson, L.E.: Ternary quadratic forms and congruences. Ann. Math. 28, 333–341 (1927)
Dickson, L.E.: Additive number theory for all quadratic functions. Am. J. Math. 50, 1–48 (1928)
Dickson, L.E.: Quadratic functions of forms, sums of whose values give all positive integers. J. Math. Pures Appl. 7, 319–336 (1928)
Dickson, L.E.: Studies in the Theory of Numbers. University of Chicago Press, Chicago (1930)
Dickson, L.E.: Polygonal numbers and related Waring problems. Q. J. Math. 5, 283–290 (1934)
Dickson, L.E.: Cyclotomy, higher congruences, and Waring’s problem, II. Am. J. Math. 57, 463–474 (1935)
Dietmann, R.: Simultaneous Diophantine approximation by square-free numbers. Q. J. Math. 59, 311–319 (2008)
Dieudonné, J.: Jacques Herbrand et la théorie des nombres. In: Proceedings of the Herbrand Symposium, Marseilles, 1981, pp. 3–7. North-Holland, Amsterdam (1982)
Dieudonné, J., Tits, J.: Claude Chevalley (1909–1984). Bull. Am. Math. Soc. 17, 1–7 (1987)
Dodson, M.: The average order of two arithmetical functions. Acta Arith. 16, 71–84 (1969/1970)
Dodson, M.: On Waring’s problem in p-adic fields. Acta Arith. 22, 315–327 (1972/1973)
Domar, Y.: On the Diophantine equation |Ax n−By n|=1, n≥5. Math. Scand. 2, 29–32 (1954)
Dong, X., Shiu, W.C., Chu, C.I., Cao, Z.: The simultaneous Pell equations y 2−Dz 2=1 and x 2−2Dz 2=1. Acta Arith. 126, 115–123 (2007)
Dörge, K.: Zur Verteilung der quadratischen Reste. Jahresber. Dtsch. Math.-Ver. 38, 41–49 (1929)
Dransfield, M.R., Liu, L., Marek, V.W., Truszczyński, M.: Satisfiability and computing van der Waerden numbers. Electron. J. Comb. 11, #R41 (2004)
Duffin, R.J., Schaeffer, A.C.: Khintchine’s problem in metric Diophantine approximation. Duke Math. J. 8, 243–255 (1941)
Duke, W., Schulze-Pillot, R.: Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 99, 49–57 (1990)
Dupré, A.: Examen d’une proposition de Legendre relative à la théorie des nombres. Mallet-Bachelier, Paris (1859)
Dusart, P.: Sur la conjecture π(x+y)≤π(x)+π(y). Acta Arith. 102, 295–308 (2002)
Dvornicich, R., Zannier, U.: Local-global divisibility of rational points in some commutative algebraic groups. Bull. Soc. Math. Fr. 129, 317–338 (2001)
Dvornicich, R., Zannier, U.: An analogue for elliptic curves of the Grunwald-Wang example. C. R. Acad. Sci. Paris 338, 47–50 (2004)
Dvornicich, R., Zannier, U.: On a local-global principle for the divisibility of a rational point by a positive integer. Bull. Lond. Math. Soc. 39, 27–34 (2007)
Dwork, B.: On the Artin root number. Am. J. Math. 78, 444–472 (1956)
Dyson, F.J.: The approximation to algebraic numbers by rationals. Acta Math. 79, 225–240 (1947)
Ehrhart, E.: Sur les polyèdres rationnels homothétiques à n dimensions. C. R. Acad. Sci. Paris 254, 616–618 (1962)
Ehrhart, E.: Sur un problème de géométrie diophantienne linéaire. I. Polyèdres et réseaux. J. Reine Angew. Math. 226, 1–29 (1967)
Ehrhart, E.: Sur un problème de géométrie diophantienne linéaire. II, Systèmes diophantiens linéaires. J. Reine Angew. Math. 227, 25–49 (1967)
Ehrhart, E.: Polynômes arithmétiques et méthode des polyèdres en combinatoire. Institut de recherche mathématique avancée, Strasbourg (1974); 2nd ed. Birkhäuser, 1977.
Eichler, M.: Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion. Arch. Math. 5, 355–366 (1954)
Eisenstein, G.: Beweis des Reciprocitätssatzes für die cubischen Reste in der Theorie der aus dritten Wurzeln der Einheit zusammengesetzten complexen Zahlen. J. Reine Angew. Math. 27, 289–310 (1844)
Eisenstein, G.: Über ein einfaches Mittel zur Auffindung der höheren Reziprozitätsgesetze und der mit ihnen verbundenen Ergänzungssätzen. J. Reine Angew. Math. 39, 351–364 (1850)
Elliott, P.D.T.A.: On the mean value of f(p). Proc. Lond. Math. Soc. 21, 28–96 (1970)
Elliott, P.D.T.A.: On the least pair of consecutive quadratic non-residues (mod p). In: Proceedings of the Number Theory Conference, Boulder, CO, 1972, pp. 75–79. University of Colorado Press, Boulder (1972)
Elsholtz, C.: The number Γ(k) in Waring’s problem. Acta Arith. 131, 43–49 (2008)
Erdős, P.: On the difference of consecutive primes. Q. J. Math. 6, 124–128 (1935)
Erdős, P.: On the representation of an integer as the sum of k k-th powers. J. Lond. Math. Soc. 11, 133–136 (1936)
Erdős, P.: Some problems and results in elementary number theory. Publ. Math. (Debr.) 2, 103–109 (1951)
Erdős, P.: On the greatest prime factor of \(\prod_{k=1}^{x}f(k)\). J. Lond. Math. Soc. 27, 379–384 (1952)
Erdős, P.: Arithmetical properties of polynomials. J. Lond. Math. Soc. 28, 416–425 (1953)
Erdős, P.: On the integers relatively prime to n and on a number-theoretic function considered by Jacobsthal. Math. Scand. 10, 163–170 (1962)
Erdős, P.: On the distribution of the convergents of almost all real numbers. J. Number Theory 2, 425–441 (1970)
Erdős, P., Graham, R.L.: Old and New Problems and Results in Combinatorial Number Theory. Université de Genève, Genève (1980)
Erdős, P., Rado, R.: Combinatorial theorems on classifications of subsets of a given set. Proc. Lond. Math. Soc. 2, 417–439 (1952)
Erdős, P., Schinzel, A.: On the greatest prime factor of \(\prod_{k=1}^{x}f(k)\). Acta Arith. 55, 191–200 (1990)
Eskin, A., Margulis, G.A., Mozes, S.: On a quantitative version of the Oppenheim conjecture. Electron. Res. Announc. Am. Math. Soc. 1, 124–130 (1995)
Eskin, A., Margulis, G.A., Mozes, S.: Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Ann. Math. 147, 93–141 (1998)
Estermann, T.: Vereinfachter Beweis eines Satzes von Kloosterman. Abh. Math. Semin. Univ. Hamb. 7, 82–98 (1929)
Estermann, T.: Proof that every large integer is a sum of seventeen biquadrates. Proc. Lond. Math. Soc. 41, 126–142 (1936)
Estermann, T.: On the representation of a number as a sum of three squares. Proc. Lond. Math. Soc. 9, 575–594 (1959)
Euler, L.: Theorematum quorundam arithmeticorum demonstrationes. Comment. Acad. Sci. Petropol. 10(1738), 125–146 (1747) [[1908], ser. 1, vol. 2, pp. 38–58]
Euler, L.: Vollständige Anleitung zur Algebra. Akademie der Wissenschaften, St. Petersbourg (1770) [[1908], ser. 1, vol. 1]
Euler, L., Goldbach, C.: Briefwechsel 1729–1764. Akademie Verlag, Berlin (1965)
Evertse, J.-H.: On the equation ax n−by n=c. Compos. Math. 47, 289–315 (1982)
Evertse, J.-H.: Upper bounds for the number of solutions of Diophantine equations. In: Math. Centre Tracts, vol. 168, pp. 1–125. Mathematisch Centrum, Amsterdam (1983)
Evertse, J.-H.: On the representation of integers by binary cubic forms of positive discriminant. Invent. Math. 73, 117–138 (1983); corr. 75, 379 (1984)
Ewing, H., Gehring, F.W. (eds.): Paul Halmos. Celebrating 50 Years of Mathematics. Springer, Berlin (1991)
Faber, G.: Über arithmetische Eigenschaften analytischer Funktionen. Math. Ann. 58, 545–557 (1904)
Faddeev, D.K.: Über die Gleichung x 4−Ay 4=±1. Tr. Mat. Inst. Steklova 5, 41–52 (1934)
Faivre, C.: Distribution of Lévy constants for quadratic numbers. Acta Arith. 61, 13–34 (1992)
Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73, 349–366 (1983); corr. vol. 75, 1984, p. 381
Feldman, N.I.: Approximation of certain transcendental numbers, I, Approximation of logarithms of algebraic numbers. Izv. Akad. Nauk SSSR, Ser. Mat. 15, 53–74 (1951) (in Russian)
Feldman, N.I.: Approximation of certain transcendental numbers, II. The approximation of certain numbers connected with the Weierstrass function ℘(z). Izv. Akad. Nauk SSSR, Ser. Mat. 15, 153–176 (1951) (in Russian)
Feldman, N.I.: The measure of transcendency of the number π. Izv. Akad. Nauk SSSR, Ser. Mat. 24, 357–368 (1960) (in Russian)
Feldman, N.I.: On the measure of transcendence of the number e. Usp. Mat. Nauk 18(3), 207–213 (1963) (in Russian)
Feldman, N.I.: An effective strengthening of Liouville’s theorem. Izv. Akad. Nauk SSSR, Ser. Mat. 35, 973–990 (1971) (in Russian)
Feldman, N.I.: The Seventh Problem of Hilbert. Moscow State University Press, Moscow (1982) (in Russian)
Fermat, P.: Observationes Domini Petri de Fermat. In: [2040], vol. 1, pp. 292–342 [French translation: [1991], vol. 3, pp. 241–274]
Fermat, P.: Letter to Mersenne, September 1636. In: [2040], vol. 2, p. 65
Filaseta, M.: Short interval results for squarefree numbers. J. Number Theory 35, 128–149 (1990)
Filaseta, M., Trifonov, O.: On gaps between squarefree numbers. Prog. Math. 85, 235–253 (1990)
Filaseta, M., Trifonov, O.: On gaps between squarefree numbers, II. J. Lond. Math. Soc. 45, 215–221 (1992)
Filaseta, M., Trifonov, O.: The distribution of fractional parts with applications to gap results in number theory. Proc. Lond. Math. Soc. 73, 241–278 (1996)
Finkelstein, R., London, H.: Completion of a table of O. Hemer. In: Proceedings of the Washington State University Conference on Number Theory, pp. 148–159. Washington State Univ., Pullman (1971) (Finkelstein, R. = Steiner, R.P.)
Fisher, T.: A counterexample to a conjecture of Selmer. In: Number Theory and Algebraic Geometry, pp. 119–131. Cambridge University Press, Cambridge (2003)
Flett, T.M.: On the function \(\sum_{n=1}^{\infty}{1\over n}\sin{t\over n}\). J. Lond. Math. Soc. 25, 5–19 (1950)
Fogels, E.: On average values of arithmetic functions. Proc. Camb. Philos. Soc. 37, 358–372 (1941)
Ford, K.B.: New estimates for mean values of Weyl sums. Internat. Math. Res. Notices, 1995, nr. 3, 155–171
Ford, K.: Waring’s problem with polynomial summands. J. Lond. Math. Soc. 61, 671–680 (2000)
Ford, K.: Vinogradov’s integral and bounds for the Riemann zeta function. Proc. Lond. Math. Soc. 85, 565–633 (2002)
Fouvry, É., Tenenbaum, G.: Entiers sans grand facteur premier en progressions arithmetiques. Proc. Lond. Math. Soc. 63, 449–494 (1991)
Frei, G.: Heinrich Weber and the emergence of class field theory. In: The History of Modern Mathematics, vol. 1, pp. 425–450. Academic Press, San Diego (1989)
Frewer, M.: Felix Bernstein. Jahresber. Dtsch. Math.-Ver. 83, 84–95 (1981)
Frobenius, G.: Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe. SBer. Preuß. Akad. Wiss. Berlin, 1896, 689–703. [[2120], vol. 2, pp. 719–733]
Fröhlich, A.: Galois Module Structure of Algebraic Integers. Springer, Berlin (1983)
Fueter, R.: Die Theorie der Zahlstrahlen. J. Reine Angew. Math. 130, 197–237 (1905)
Fueter, R.: Die Theorie der Zahlstrahlen, II. J. Reine Angew. Math. 132, 255–269 (1907)
Fueter, R.: Die Klassenanzahl der Körper der komplexen Multiplikation. Nachr. Ges. Wiss. Göttingen, 1907, 288–298
Fueter, R.: Die verallgemeinerte Kroneckersche Grenzformel und ihre Anwendung auf die Berechnung der Klassenzahlen. Rend. Circ. Mat. Palermo 29, 380–395 (1910)
Fueter, R.: Die Klassenkörper der komplexen Multiplikation und ihr Einfluss auf die Entwicklung der Zahlentheorie. Jahresber. Dtsch. Math.-Ver. 20, 1–47 (1911) [Reprint: Teubner, 1958]
Fueter, R.: Abelsche Gleichungen in quadratisch-imaginären Zahlkörpern. Math. Ann. 75, 177–255 (1914)
Fueter, R.: Vorlesungen über die singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen. Teubner, Leipzig (1924–1927)
Fueter, R.: Ueber kubische diophantische Gleichungen. Comment. Math. Helv. 2, 69–89 (1930)
Fujiwara, M.: Hasse principle in algebraic equations. Acta Arith. 22, 267–276 (1972)
Fujiwara, M., Sudo, M.: Some forms of odd degree for which the Hasse principle fails. Pac. J. Math. 67, 161–169 (1976)
Furstenberg, H., Weiss, B.: Markov processes and Ramsey theory for trees. Comb. Probab. Comput. 12, 547–563 (2003)
Furtwängler, Ph.: Über das Reciprocitätsgesetz der l ten Potenzreste in algebraischen Zahlkörpern, wenn l eine ungerade Primzahl bedeutet. Abhandl. Ges. Wiss. Göttingen Math. Phys. Kl. 2(3), 3–82 (1902)
Furtwängler, Ph.: Die Konstruktion des Klassenkörpers für solche algebraische Zahlkörper, die eine lte Einheitswurzel enthalten und deren Idealklassen eine cyclische Gruppe vom Grade l h bilden. Nachr. Ges. Wiss. Göttingen, 1903, 203–217
Furtwängler, Ph.: Ueber die Konstruktion des Klassenkörpers für beliebige algebraische Zahlkörper, die eine lte Einheitswurzel enthalten. Nachr. Ges. Wiss. Göttingen, 1903, 282–303
Furtwängler, Ph.: Die Konstruktion des Klassenkörpers für beliebige algebraische Zahlkörper. Nachr. Ges. Wiss. Göttingen, 1904, 173–195
Furtwängler, Ph.: Über die Reziprozitätsgesetze zwischen l-ten Potenzresten in algebraischen Zahlkörpern, wenn l eine ungerade Primzahl bedeutet. Math. Ann. 58, 1–50 (1904)
Furtwängler, Ph.: Eine charakteristische Eigenschaft des Klassenkörpers. Nachr. Ges. Wiss. Göttingen, 1906, 417–434; 1907, 1–24
Furtwängler, P.: Allgemeiner Existenzbeweis für den Klassenkörper eines beliebigen algebraischen Zahlkörpers. Math. Ann. 63, 1–37 (1907)
Furtwängler, P.: Die Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern, I. Math. Ann. 67, 1–31 (1909)
Furtwängler, P.: Die Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern, II. Math. Ann. 72, 346–386 (1912)
Furtwängler, P.: Die Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern, III. Math. Ann. 74, 413–429 (1913)
Furtwängler, P.: Allgemeiner Beweis des Zerlegungssatzes für den Klassenkörper. Nachr. Ges. Wiss. Göttingen, 1911, 293–317
Furtwängler, P.: Über das Verhalten der Ideale des Grundkörpers im Klassenkörper. Monatshefte Math. Phys. 27, 1–15 (1916)
Furtwängler, P.: Über die Reziprozitätgesetze für Primzahlpotenzexponente. J. Reine Angew. Math. 157, 15–25 (1927)
Furtwängler, P.: Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper. Abh. Math. Semin. Univ. Hamb. 7, 14–36 (1930)
Gajda, W., Górnisiewicz, K.: Linear dependence in Mordell-Weil groups. J. Reine Angew. Math. 630, 219–233 (2009)
Gajraj, J.: Simultaneous approximation to certain polynomials. J. Lond. Math. Soc. 14, 527–534 (1976)
Gallagher, P.[X.]: Approximation by reduced fractions. J. Math. Soc. Jpn. 13, 342–345 (1961)
Gatteschi, L.: Un perfezionamento di un teorema di I. Schur sulla frequenza dei numeri primi. Boll. Unione Mat. Ital. 2, 123–125 (1947)
Gauss, C.F.: Letter to Laplace, 30.01.1812. In: [2263], vol. 101, pp. 371–374
Gauss, C.F.: Theoria residuorum biquadraticorum. Comm. Soc. Reg. Sci. Gottingensis, 6 (1828); 7 (1832) [[2214], vol. 2, pp. 65–92, 93–148]
Gebel, J., Pethő, A., Zimmer, H.G.: On Mordell’s equation. Compos. Math. 110, 335–367 (1998)
Gelbart, S.: Automorphic forms and Artin’s conjecture. In: Lecture Notes in Math., vol. 627, pp. 241–276. Springer, Berlin (1977)
Gelfond, A.O.: Approximation of algebraic irrationalities and their logarithms. Vestn. Mosk. Univ. 9, 3–25 (1948) (in Russian) [[2231], pp. 129–150]
Good, A.: Ein Ω-Resultat für das quadratische Mittel der Riemannscher Zetafunktion auf der kritischen Linie. Invent. Math. 41, 233–251 (1977)
Good, I.J.: The fractional dimensional theory of continued fractions. Proc. Camb. Philos. Soc. 37, 199–228 (1941)
Götzky, F.: Über eine zahlentheoretische Anwendung von Modulfunktionen zweier Veränderlichen. Math. Ann. 100, 411–437 (1928)
Gourdon, X.: The 1013 first zeros of the Riemann zeta function, and zeros computation at very large height. Preprint (2004)
Gowers, W.T.: A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11, 465–588 (2001); err.: p. 869
Graham, R.L.: On quadruples of consecutive kth power residues. Proc. Am. Math. Soc. 15, 196–197 (1964)
Graham, R.L., Rothschild, B.L.: A short proof of van der Waerden’s theorem on arithmetic progression. Proc. Am. Math. Soc. 42, 385–386 (1974)
Graham, S.W.: An algorithm for computing optimal exponent pairs. J. Lond. Math. Soc. 33, 203–218 (1986)
Graham, S.W., Kolesnik, G.: On the difference between consecutive squarefree integers. Acta Arith. 49, 435–447 (1988)
Graham, S.W., Kolesnik, G.: Van der Corput’s Method of Exponential Sums. Cambridge University Press, Cambridge (1991)
Granville, A.: Integers, without large prime factors, in arithmetic progressions, I. Acta Math. 170, 255–273 (1993)
Granville, A.: Integers, without large prime factors, in arithmetic progressions, II. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 345, 349–362 (1993)
Granville, A.: ABC allows us to count squarefrees. Internat. Math. Res. Notices, 1998, 991–1009
Greminger, H.: Sur le nombre e. Enseign. Math. 29, 255–259 (1930)
Grimshaw, M.E.: Hans Ludwig Hamburger. J. Lond. Math. Soc. 33, 377–383 (1958)
Grošev, A.V.: Un théorème sur les systemes de formes linéaires. Dokl. Akad. Nauk SSSR 19, 151–152 (1938)
Gross, B.H.: Arithmetic on Elliptic Curves with Complex Multiplication. Lecture Notes in Math., vol. 776. Springer, Berlin (1980)
Grosswald, E.: Representations of Integers as Sums of Squares. Springer, Berlin (1985)
Gruenberger, F., Armerding, G.: Statistics of the first six million primes. Report P-2460, The Rand Corporation (1961)
Hafner, J.L., Ivić, A.: On the mean-square of the Riemann zeta-function on the critical line. J. Number Theory 32, 151–191 (1989)
Hagedorn, T.: Computation of Jacobsthal’s function h(n) for n<50. Math. Comput. 78, 1073–1087 (2009)
Halmos, P.: Note on almost-universal forms. Bull. Am. Math. Soc. 44, 141–144 (1938)
Halupczok, K.: On the number of representations in the ternary Goldbach problem with one prime number in a given residue class. J. Number Theory 117, 292–300 (2006)
Halupczok, K.: Zum ternären Goldbachproblem mit Kongruenzbedingungen an die Primzahlen. In: Elementary and Analytic Number Theory. Proceedings of the ELAZ conference, Stuttgart, May 24–28, 2004, pp. 57–64. Franz Steiner Verlag, Stuttgart (2006)
Hamburger, H.: Über die Riemannsche Funktionalgleichung der ζ-Funktion. SBer. Berl. Math. Ges. 20, 4–9 (1921)
Hamburger, H.: Über die Funktionalgleichung der L-Reihen. SBer. Berl. Math. Ges. 20, 10–13 (1921)
Hamburger, H.: Über die Riemannsche Funktionalgleichung der ζ-Funktion, I. Math. Z. 10, 240–254 (1921)
Hamburger, H.: Über die Riemannsche Funktionalgleichung der ζ-Funktion, II. Math. Z. 11, 224–245 (1922)
Hamburger, H.: Über die Riemannsche Funktionalgleichung der ζ-Funktion, III. Math. Z. 13, 283–311 (1922)
Hamburger, H.: Bemerkungen zu einem Satze über die Riemannsche ζ Funktion. SBer. Bayer. Akad. Wiss., 1922, 151–156
Hančl, J.: Two proofs of transcendency of π and e. Czechoslov. Math. J. 35, 543–549 (1985)
Haneke, W.: Verschärfung der Abschätzung von ζ(1/2+it). Acta Arith. 8, 357–430 (1962/1963)
Hardy, G.H.: On the representation of a number as the sum of any number of squares, and in particular of five. Trans. Am. Math. Soc. 21, 255–284 (1920)
Hardy, G.H.: Divergent Series. Oxford University Press, Oxford (1949)
Hardy, G.H., Littlewood, J.E.: Some problems of diophantine approximation. In: Proceedings of the 5th ICM, pp. 223–229. Cambridge University Press, Cambridge (1912)
Hardy, G.H., Littlewood, J.E.: Some problems of diophantine approximation, I. The fractional part of n kΘ. Acta Math. 37, 155–191 (1914)
Hardy, G.H., Littlewood, J.E.: Some problems of diophantine approximation, II. The trigonometrical series associated with the elliptic ϑ-functions. Acta Math. 37, 193–238 (1914)
Hardy, G.H., Littlewood, J.E.: Contributions to the theory of Riemann zeta-function and the theory of distribution of primes. Acta Math. 41, 119–196 (1917)
Hardy, G.H., Littlewood, J.E.: A new solution of Waring’s problem. Q. J. Math. 48, 272–293 (1919)
Hardy, G.H., Littlewood, J.E.: Some problems of ‘Partitio Numerorum’; I: A new solution of Waring’s problem. Nachr. Ges. Wiss. Göttingen, 1920, 33–54
Hardy, G.H., Littlewood, J.E.: Some problems of ‘Partitio Numerorum’: II. Proof that every large number is a sum of at most 21 biquadrates. Math. Z. 9, 14–27 (1921)
Hardy, G.H., Littlewood, J.E.: The zeros of Riemann’s zeta-function on the critical line. Math. Z. 10, 283–317 (1921)
Hardy, G.H., Littlewood, J.E.: Some problems of diophantine approximation: the lattice-points in a right-angled triangle. Proc. Lond. Math. Soc. 20, 15–36 (1922)
Hardy, G.H., Littlewood, J.E.: Some problems of diophantine approximation: the lattice-points in a right-angled triangle, II. Abh. Math. Semin. Univ. Hamburg 1, 212–2496 (1922)
Hardy, G.H., Littlewood, J.E.: The approximate functional equation in the theory of zeta-function, with applications to the divisor-problems of Dirichlet and Piltz. Proc. Lond. Math. Soc. 21, 39–74 (1923)
Hardy, G.H., Littlewood, J.E.: Some problems of ‘Partitio Numerorum’: III. On the expression of a number as a sum of primes. Acta Math. 44, 1–70 (1923)
Hardy, G.H., Littlewood, J.E.: Some problems of ‘Partitio Numerorum’: IV. The singular series in Waring’s problem and the value of the number G(k). Math. Z. 12, 161–188 (1922)
Hardy, G.H., Littlewood, J.E.: On Lindelöf’s hypothesis concerning the Riemann zeta-function. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 103, 403–412 (1923)
Hardy, G.H., Littlewood, J.E.: Some problems of ‘Partitio Numerorum’ (VI): Further researches in Waring’s problem. Math. Z. 23, 1–37 (1925)
Hardy, G.H., Littlewood, J.E.: Some problems of ‘Partitio Numerorum’ (VIII): The number Γ(k) in Waring’s problem. Proc. Lond. Math. Soc. 28, 518–542 (1928)
Hardy, G.H., Littlewood, J.E.: The approximate functional equations for ζ(s) and ζ2(s). Proc. Lond. Math. Soc. 29, 81–97 (1929)
Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 17, 75–115 (1918) [[5088], pp. 276–309]
Harman, G.: Primes in short intervals. Math. Z. 180, 335–348 (1982)
Harman, G.: Diophantine approximation with square-free integers. Math. Proc. Camb. Philos. Soc. 95, 381–388 (1984)
Harman, G.: Metric Diophantine approximation with two restricted variables. I. Two square-free integers, or integers in arithmetic progressions. Proc. Camb. Philos. Soc. 103, 197–206 (1988)
Harman, G.: Some cases of the Duffin and Schaeffer conjecture. Q. J. Math. 41, 395–404 (1990)
Harman, G.: Metric Number Theory. Oxford University Press, Oxford (1998)
Harman, G.: Integers without large prime factors in short intervals and arithmetic progressions. Acta Arith. 91, 279–289 (1999)
Haselgrove, C.B., Miller, J.C.P.: Tables of the Riemann Zeta Function. Cambridge University Press, Cambridge (1960)
Hasse, H.: Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen. J. Reine Angew. Math. 152, 129–148 (1923) [[2607], vol. 1, pp. 3–42]
Hasse, H.: Über die Äquivalenz quadratischer Formen im Körper der rationalen Zahlen. J. Reine Angew. Math. 152, 205–224 (1923) [[2607], vol. 1, pp. 23–42]
Hasse, H.: Zur Theorie des quadratischen Hilbertschen Normenrestsymbols in algebraischen Körpern. J. Reine Angew. Math. 153, 76–93 (1923)
Hasse, H.: Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper. J. Reine Angew. Math. 153, 113–130 (1924) [[2607], vol. 1, pp. 75–92]
Hasse, H.: Äquivalenz quadratischer Formen in einem beliebigen algebraischen Zahlkörper. J. Reine Angew. Math. 153, 158–162 (1924) [[2607], vol. 1, pp. 93–97]
Hasse, H.: Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, I. Jahresber. Dtsch. Math.-Ver. 35, 1–55 (1926) [2nd ed. 1965, 3rd ed. 1970, Physica Verlag]
Hasse, H.: Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Ia. Jahresber. Dtsch. Math.-Ver. 36, 233–311 (1927) [2nd ed. 1965, 3rd ed. 1970, Physica Verlag]
Hasse, H.: Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, II. In: Jber. Deutsch. Math.-Verein. Ergänzungsband, vol. 6, pp. 1–204. Teubner, Leipzig (1930) [2nd ed. 1965, 3rd ed. 1970, Physica Verlag]
Hasse, H.: Das Eisensteinsche Reziprozitätsgesetz der n-ten Potenzreste. Math. Ann. 97, 599–623 (1927) [[2607], vol. 1, pp. 269–293]
Hasse, H.: Über das Reziprozitätsgesetz der m-ten Potenzreste. J. Reine Angew. Math. 158, 228–259 (1927) [[2607], vol. 1, pp. 294–325]
Hasse, H.: Neue Begründung der komplexen Multiplikation, I. J. Reine Angew. Math. 157, 115–139 (1927) [[2607], vol. 2, pp. 3–27]
Hasse, H.: Neue Begründung der komplexen Multiplikation, II. J. Reine Angew. Math. 165, 64–88 (1931) [[2607], vol. 2, pp. 28–52]
Hasse, H.: Zum expliziten Reziprozitätsgesetz. Abh. Math. Semin. Univ. Hamb. 7, 52–63 (1929) [[2607], vol. 1, pp. 343–354]
Hasse, H.: Neue Begründung und Verallgemeinerung der Theorie des Normenrestsymbols. J. Reine Angew. Math. 162, 134–144 (1930) [[2607], vol. 1, pp. 134–144Footnote
The same pagination occurring in two places is not a result of a printing error, but occurred really.
]Hasse, H.: Die Normenresttheorie relativ-abelscher Zahlkörper als Klassenkörpertheorie im Kleinen. J. Reine Angew. Math. 162, 145–154 (1930) [[2607], vol. 1, pp. 145–154]
Hasse, H.: Beweis eines Satzes und Widerlegung einer Vermutung über das allgemeine Normenrestsymbol. Nachr. Ges. Wiss. Göttingen, 64–69 (1931) [[2607], vol. 1, pp. 155–160]
Hasse, H.: Die Struktur der R. Brauerschen Algebrenklassengruppe über einem algebraischen Zahlkörper. Math. Ann. 107, 731–760 (1933) [[2607], vol. 1, pp. 501–530]
Hasse, H.: Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper. J. Reine Angew. Math. 172, 37–54 (1935) [[2607], vol. 2, pp. 133–150]
Hasse, H.: Der n-Teilungskörper eines abstrakten elliptischen Funktionenkörpers als Klassenkörper, nebst Anwendung auf den Mordell-Weilschen Endlichkeitssatz. Math. Z. 48, 48–66 (1942) [[2607], vol. 2, pp. 403–421]
Hasse, H.: Rein-arithmetischer Beweis des Siegelschen Endlichkeitssatzes für binäre diophantische Gleichungen im Spezialfall des Geschlechts 1. Abhandl. Deutsch. Akad. Wiss., 1951, nr. 2, 1–19
Hasse, H., Hensel, K.: Über die Normenreste eines relativ-zyklischen Körpers vom Primzahlgrad l nach einem Primteiler \(\mathfrak{l}\) von l. Math. Ann. 90, 262–278 (1923) [[2607], vol. 1, pp. 101–117]
Hata, M.: Rational approximations to π and some other numbers. Acta Arith., 1993, 335–349
Haussner, R.: Tafeln für das Goldbach’sche Gesetz. Abhandlungen der Kais. Leopold. Carol. Deutsch. Akad. der Naturforscher, Halle 1, 1–214 (1899)
Hazewinkel, M.: Local class field theory is easy. Adv. Math. 18, 148–181 (1975)
Heath-Brown, D.R.: The fourth power moment of the Riemann zeta function. Proc. Lond. Math. Soc. 38, 385–422 (1979)
Heath-Brown, D.R.: Fractional moments of the Riemann zeta-function. J. Lond. Math. Soc. 24(2), 65–78 (1981)
Heath-Brown, D.R.: Gaps between primes, and the pair correlation of zeros of the zeta function. Acta Arith. 41, 85–99 (1982)
Heath-Brown, D.R.: Primes in “almost all” short intervals. J. Lond. Math. Soc. 26, 385–396 (1982)
Heath-Brown, D.R.: Diophantine approximation with square-free numbers. Math. Z. 187, 335–344 (1984)
Heath-Brown, D.R.: Weyl’s inequality, Hua’s inequality and Waring’s problem. J. Lond. Math. Soc. 38, 216–230 (1988)
Heath-Brown, D.R.: The fractional part of αn k. Mathematika 35, 28–37 (1988)
Heath-Brown, D.R.: The largest prime factor of X 3+2. Proc. Lond. Math. Soc. 82, 554–596 (2001)
Heath-Brown, D.R.: Primes represented by x 3+2y 3. Acta Math. 186, 1–84 (2001)
Heath-Brown, D.R.: The density of rational points on curves and surfaces. Ann. Math. 155, 553–595 (2002)
Heath-Brown, D.R., Goldston, D.A.: A note on the differences between consecutive primes. Math. Ann. 266, 317–320 (1984)
Heath-Brown, D.R., Huxley, M.N.: Exponential sums with a difference. Proc. Lond. Math. Soc. 61, 227–250 (1990)
Heath-Brown, D.R., Iwaniec, H.: On the difference between consecutive prime numbers. Invent. Math., 55, 49–69
Hecke, E.: Reziprozitätsgesetz und Gauss’sche Summen in quadratischen Zahlkörpern. Nachr. Ges. Wiss. Göttingen, 1919, 265–278 [[2703], pp. 235–248]
Hecke, E.: Über die Lösung der Riemannscher Funktionalgleichung. Math. Z. 16, 301–307 (1923) [[2703], pp. 374–380]
Hecke, E.: Vorlesungen über die Theorie der algebraischen Zahlen. Akademische Verlag, Leipzig (1923) [Reprint: Chelsea, 1948; 2nd ed., Leipzig, 1954; reprint: Chelsea, 1970; English translation: Lectures on the Theory of Algebraic Numbers, Springer, 1981]
Hecke, E.: Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen. Math. Ann. 112, 664–699 (1936) [[2703], pp. 591–626]
Heilbronn, H.: Über den Primzahlsatz von Herrn Hoheisel. Math. Z., 36, 394–423. [[2715], pp. 70–99]
Heilbronn, H.: On the distribution of the sequence n 2 θ mod 1. Q. J. Math. 19, 249–256 (1948)
Hemer, O.: On the Diophantine equation y 2−k=x 3. Dissertation, Univ. of Uppsala (1952)
Hemer, O.: Notes on the Diophantine equation y 2−k=x 3. Ark. Mat. 3, 67–77 (1954)
Hensley, D.: The Hausdorff dimensions of some continued fraction Cantor sets. J. Number Theory 33, 182–198 (1989)
Hensley, D.: Continued fraction Cantor sets, Hausdorff dimension, and functional analysis. J. Number Theory 40, 336–358 (1992)
Hensley, D.: A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets. J. Number Theory 58, 9–45 (1996); corr. 59, 419 (1996)
Hensley, D., Richards, I.: On the incompatibility of two conjectures concerning primes. In: Proc. Symposia Pure Math., vol. 24, pp. 123–127. Am. Math. Soc., Providence (1973)
Hensley, D., Richards, I.: Primes in intervals. Acta Arith. 25, 375–391 (1974)
Herbrand, J.: Théorie arithmétique des corps de nombres à degré infini. Math. Ann. 106, 473–501 (1932)
Herbrand, J.: Théorie arithmétique des corps de nombres à degré infini, II. Math. Ann. 108, 699–717 (1933)
Hermite, C.: Sur la fonction exponentielle. C. R. Acad. Sci. Paris 77, 18–24, 74–79, 221–233, 285–293 (1873) [[2766], vol. 3, pp. 150–181]
Hilbert, D.: Ueber die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann. 32, 342–350 (1888) [[2792], vol. 2, pp. 154–161]
Hilbert, D.: Die Theorie der algebraischer Zahlkörper. Jahresber. Dtsch. Math.-Ver. 4, 175–546 (1897) [[2792], 1, 63–363; English translation: The Theory of Algebraic Number Fields, Springer, 1998; French translation: Ann. Fac. Sci. Toulouse 1, 257–328 (1909); 2, 225–456 (1910)]
Hilbert, D.: Über Diophantische Gleichungen. Nachr. Ges. Wiss. Göttingen, 1897, 48–54 [[2792], vol. 2, pp. 384–389]
Hilbert, D.: Ueber die Theorie der relativ-Abelschen Zahlkörper Nachr. Ges. Wiss. Göttingen, 1898, 377–399. Acta Math. 26, 1902, 99–132 [[2792], vol. 1, pp. 483–509]
Hilbert, D.: Ueber die Theorie des relativ-quadratischen Zahlkörpers. Math. Ann. 51, 1–127 (1899) [[2792], vol. 1, pp. 370–482]
Hilbert, D.: Grundlagen der Geometrie. Teubner, Leipzig (1899); 2nd ed. 1903, 3rd ed. 1909, …, 14th ed. 1999
Hildebrand, A.: On the number of positive integers ≤x and free of prime factors >y. J. Number Theory 22, 289–307 (1986)
Hildebrand, A.: On consecutive kth power residues. Monatshefte Math. 102, 103–114 (1986)
Hildebrand, A.: On consecutive kth power residues, II. Mich. Math. J. 39, 241–253 (1991)
Hildebrand, A.: On the least pair of consecutive quadratic nonresidues. Mich. Math. J. 34, 57–62 (1987)
Hildebrand, A., Tenenbaum, G.: On integers free of large prime factors. Trans. Am. Math. Soc. 296, 265–290 (1986)
Hildebrand, A., Tenenbaum, G.: Integers without large prime factors. J. Théor. Nr. Bordx. 5, 411–484 (1993)
Hilton, P.J.: Heinz Hopf. Bull. Lond. Math. Soc. 4, 202–217 (1972)
Hlawka, E.: Zur formalen Theorie der Gleichverteilung in kompakten Gruppen. Rend. Circ. Mat. Palermo 4, 33–47 (1955)
Hoheisel, G.: Primzahlprobleme in der Analysis. SBer. Preuß. Akad. Wiss. Berlin, 1930, 580–588
Holzer, L.: Minimal solutions of diophantine equations. Can. J. Math. 2, 238–244 (1950)
Hooley, C.: On the representation of a number as the sum of two squares and a prime. Acta Math. 97, 189–210 (1957)
Hooley, C.: On the greatest prime factor of a quadratic polynomial. Acta Math. 117, 281–299 (1967)
Hooley, C.: On the power free values of polynomials. Mathematika 14, 21–26 (1967)
Hooley, C.: On the distribution of square-free numbers. Can. J. Math. 25, 1216–1223 (1973)
Hooley, C.: Applications of Sieve Methods. Academic Press, San Diego (1974)
Hooley, C.: On power-free numbers and polynomials, I. J. Reine Angew. Math. 293/294, 67–85 (1977)
Hooley, C.: On power-free numbers and polynomials, II. J. Reine Angew. Math. 295, 1–21 (1977)
Hooley, C.: On the greatest prime factor of a cubic polynomial. J. Reine Angew. Math. 303/304, 21–50 (1978)
Hooley, C.: On a problem of Hardy and Littlewood. Acta Arith. 79, 289–311 (1997)
Hopf, H.: Über die Verteilung quadratischer Reste. Math. Z. 32, 222–231 (1930)
Hsia, J.S., Icaza, M.I.: Effective version of Tartakowsky’s theorem. Acta Arith. 89, 235–253 (1999)
Hua, L.K.: On Waring’s problem with polynomial summands. Am. J. Math. 58, 553–562 (1936)
Hua, L.K.: An easier Waring-Kamke problem. J. Lond. Math. Soc. 11, 4–5 (1936)
Hua, L.K.: On a generalized Waring problem. Proc. Lond. Math. Soc. 43, 161–182 (1937) [[2937], pp. 17–38]
Hua, L.K.: On a generalized Waring problem, II. J. Chin. Math. Soc. 2, 175–191 (1940) [[2937], pp. 61–73]
Hua, L.K.: On Waring’s problem. Q. J. Math. 9, 199–202 (1938) [[2937], pp. 39–42]
Hua, L.K.: Sur le problème de Waring relatif à un polynome du troisième degré. C. R. Acad. Sci. Paris 210, 650–652 (1940)
Hua, L.K.: On a system of diophantine equations. Dokl. Akad. Nauk SSSR 27, 312–313 (1940)
Hua, L.K.: Additive Theory of Prime Numbers. Tr. Mat. Inst. Steklova 22, 1–197 (1947) (in Russian) [English translation: Am. Math. Soc., 1965; German translation: Additive Zahlentheorie, Leipzig, 1959]
Hua, L.K.: Some results on additive theory of numbers. Proc. Natl. Acad. Sci. USA 33, 136–137 (1947)
Hua, L.K.: An improvement of Vinogradov’s mean-value theorem and several applications. Q. J. Math. 20, 48–61 (1949) [[2937], pp. 178–191]
Hudson, R.H.: On sequences of consecutive quadratic non-residues. J. Number Theory 3, 178–181 (1971)
Hudson, R.H.: On a conjecture of Issai Schur. J. Reine Angew. Math. 289, 215–220 (1977)
Hughes, J.F., Shallit, J.O.: On the number of multiplicative partitions. Am. Math. Mon. 90, 468–471 (1983)
Hummel, P.: On consecutive quadratic non-residues: a conjecture of Issai Schur. J. Number Theory 103, 257–266 (2003)
Humphreys, M.G.: On the Waring problem with polynomial summands. Duke Math. J. 1, 361–375 (1935)
Hurwitz, A.: Über die Theorie der elliptischen Modulfunktionen. Math. Ann. 58, 343–360 (1904) [[2965], vol. 2, pp. 577–595]
Husemöller, D.H.: Elliptic Curves. Springer, Berlin (1987); 2nd ed. 2004
Hutchinson, J.I.: On the roots of the Riemann zeta function. Trans. Am. Math. Soc. 27, 49–60 (1925)
Huxley, M.N.: On the difference between consecutive primes. Invent. Math. 15, 164–170 (1972)
Huxley, M.N.: Exponential sums and lattice points, III. Proc. Lond. Math. Soc. 87, 591–609 (2003)
Huxley, M.N.: Exponential sums and the Riemann zeta function, IV. Proc. Lond. Math. Soc. 66, 1–40 (1993)
Huxley, M.N.: Exponential sums and the Riemann zeta function, V. Proc. Lond. Math. Soc. 90, 1–41 (2005)
Huxley, M.N.: Area, Lattice Points and Exponential Sums. Oxford University Press, Oxford (1996)
Huxley, M.N.: Moments of differences between square-free numbers. In: Sieve Methods, Exponential Sums, and Their Applications in Number Theory, Cardiff, 1995, pp. 187–204. Cambridge University Press, Cambridge (1997)
Huxley, M.N.: The rational points close to a curve, II. Acta Arith. 93, 201–219 (2000)
Huxley, M.N., Kolesnik, G.: Exponential sums and the Riemann zeta function, III. Proc. Lond. Math. Soc. 62, 449–468 (1991); corr. 66, 302 (1993)
Huxley, M.N., Nair, M.: Power free values of polynomials, III. Proc. Lond. Math. Soc. 41, 66–82 (1980)
Huxley, M.N., Watt, N.: Exponential sums and the Riemann zeta function. Proc. Lond. Math. Soc. 57, 1–24 (1988)
Hyyrö, S.: Über die Gleichung ax n−by n=z und das Catalansche Problem. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 355, 1–50 (1964)
Ingham, A.E.: Mean-value theorems in the theory of Riemann zeta-function. Proc. Lond. Math. Soc. 27, 273–300 (1928)
Iseki, K.: A remark on the Goldbach-Vinogradov theorem. Proc. Jpn. Acad. Sci. 25, 185–187 (1949)
Ishida, M.: On the divisibility of Dedekind’s zeta-functions. Proc. Jpn. Acad. Sci. 33, 293–297 (1957)
Ishida, M.-N.: Polyhedral Laurent series and Brion’s equalities. Int. J. Math. 1, 251–265 (1990)
Iskovskih, V.A.: A counterexample to the Hasse principle for systems of two quadratic forms in five variables. Mat. Zametki 10, 253–257 (1971) (in Russian)
Ivić, A.: The Riemann Zeta-function. Wiley, New York (1985) [Reprint: Dover 2003]
Ivić, A.: Lectures on Mean Values of the Riemann Zeta Function. Springer, Berlin (1991)
Ivić, A., Motohashi, Y.: The mean square of the error term for the fourth power moment of the zeta-function. Proc. Lond. Math. Soc. 69, 309–329 (1994)
Ivić, A., Motohashi, Y.: On the fourth power moment of the Riemann zeta-function. J. Number Theory 51, 16–45 (1995)
Iwaniec, H.: On the error term in the linear sieve. Acta Arith. 19, 1–30 (1971)
Iwaniec, H.: On the problem of Jacobsthal. Demonstr. Math. 11, 225–231 (1978)
Iwaniec, H., Jutila, M.: Primes in short intervals. Ark. Mat. 17, 167–176 (1979)
Iwaniec, H., Kowalski, E.: Analytic Number Theory. Am. Math. Soc., Providence (2004)
Iwaniec, H., Mozzochi, C.J.: On the divisor and circle problems. J. Number Theory 29, 60–93 (1988)
Iwaniec, H., Pintz, J.: Primes in short intervals. Monatshefte Math. 98, 115–143 (1984)
Iyanaga, S.: Zum Beweis des Hauptidealsatzes. Abh. Math. Semin. Univ. Hamb. 10, 349–357 (1934)
Iyanaga, S.: Travaux de Claude Chevalley sur la théorie du corps de classes: introduction. Jpn. J. Math. 1, 25–85 (2006)
Jacobi, C.G.J.: De usu legitimo formulae summatoriae Maclaurinianae. J. Reine Angew. Math. 12, 263–272 (1834) [[3082], vol. 6, pp. 64–75]
Jacobi, C.G.J.: De usu theoriae integralium ellipticorum et integralium abelianorum in analysi diophantea. J. Reine Angew. Math. 13, 353–355 (1835) [[3082], vol. 2, pp. 51–55]
Jacobsthal, E.: Anwendungen einer Formel aus der Theorie der quadratischen Reste. Dissertation, Univ. Berlin (1906)
Jacobsthal, E.: Über Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist, I. Norske Vid. Selsk. Forh., Trondheim 33, 117–124 (1961)
Jacobsthal, E.: Über Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist, II. Norske Vid. Selsk. Forh., Trondheim 33, 125–131 (1961)
Jacobsthal, E.: Über Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist, III. Norske Vid. Selsk. Forh., Trondheim 33, 132–139 (1961)
Jacobsthal, E.: Über Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist, IV. Norske Vid. Selsk. Forh., Trondheim 34, 1–7 (1961)
Jacobsthal, E.: Über Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist, V. Norske Vid. Selsk. Forh., Trondheim 34, 110–115 (1961)
Jager, H., Lekkerkerker, C.G.: In memoriam Prof. Dr. J. Popken. Nieuw Arch. Wiskd. 19, 1–9 (1971)
Jager, H., Liardet, P.: Distributions arithmétiques des dénominateurs de convergents de fractions continues. Indag. Math. 50, 181–197 (1988)
Jagy, W.C.: Five regular or nearly-regular ternary quadratic forms. Acta Arith. 77, 361–367 (1996)
Jagy, W.C., Kaplansky, I., Schiemann, A.: There are 913 regular ternary forms. Mathematika 44, 332–341 (1997)
James, R.D.: The representation of integers as sums of pyramidal numbers. Math. Ann. 109, 196–199 (1933)
James, R.D.: The representation of integers as sums of values of cubic polynomials. Am. J. Math. 56, 303–315 (1934)
James, R.D.: The representation of integers as sums of values of cubic polynomials, II. Am. J. Math. 59, 393–398 (1937)
Jarník, V.: Über die Gitterpunkte auf konvexen Kurven. Math. Z. 24, 500–518 (1925)
Jarník, V.: Diophantische Approximationen und Hausdorffsches Mass. Mat. Sb. 36, 371–382 (1929)
Jarník, V.: Przyczynek do metrycznej teorji przybliżeń diofantowych. Pr. Mat.-Fiz. 36, 91–106 (1929)
Jarník, V., Landau, E.: Untersuchungen über einen van der Corputschen Satz. Math. Z. 39, 745–767 (1935) [[3680], vol. 9, pp. 327–349]
Jehne, W.: On knots in algebraic number theory. J. Reine Angew. Math. 311/312, 215–254 (1979)
Jenkinson, O.: On the density of Hausdorff dimensions of bounded type continued fraction sets: the Texan conjecture. Stoch. Dyn. 4, 63–76 (2004)
Jia, C.H.: Difference between consecutive primes. Sci. China Ser. A 38, 1163–1186 (1995)
Jia, C.H.: Almost all short intervals containing prime numbers. Acta Arith. 76, 21–84 (1996)
Jones, B.W.: The regularity of a genus of positive ternary quadratic forms. Trans. Am. Math. Soc. 33, 111–124 (1931)
Jones, B.W., Pall, G.: Regular and semi-regular positive ternary quadratic forms. Acta Math. 70, 165–191 (1939)
Jones, H.: Khinchin’s theorem in k dimensions with prime numerator and denominator. Acta Arith. 99, 205–225 (2001)
Jorgenson, J., Krantz, S.G.: Serge Lang, 1927–2005. Not. Am. Math. Soc. 53, 536–553 (2006)
Jorgenson, J., Krantz, S.G.: The mathematical contributions of Serge Lang. Not. Am. Math. Soc. 54, 476–497 (2007)
Juel, C.S.: Ueber die Parameterbestimmung von Punkten auf Curven zweiter und dritter Ordnung. Eine geometrische Einleitung in die Theorie der logarithmischen und elliptischen Funktionen. Math. Ann. 47, 72–104 (1896)
Jutila, M.: Riemann’s zeta function and the divisor problem. Ark. Mat. 21, 75–96 (1983)
Jutila, M.: Riemann’s zeta function and the divisor problem, II. Ark. Mat. 31, 61–70 (1993)
Kadiri, H.: Short effective intervals containing primes in arithmetic progressions and the seven cubes problem. Math. Comput. 77, 1733–1748 (2008)
Kallies, J.: Verallgemeinerte Dedekindsche Summen und ein Gitterpunktproblem im n-dimensionalen Raum. J. Reine Angew. Math. 344, 22–37 (1983)
Kamke, E.: Verallgemeinerung des Waring-Hilbertschen Satzes. Math. Ann. 83, 85–112 (1921)
Kamke, E.: Zum Waringschen Problem für rationale Zahlen und Polynome. Math. Ann. 87, 238–245 (1922)
Kamke, E.: Über die Zerfällung rationaler Zahlen in rationale Polynomwerte. Math. Z. 12, 323–328 (1922)
Kamke, E.: Bemerkung zum allgemeinen Waringschen Problem. Math. Z. 15, 188–194 (1922)
Kamke, E.: Zur Arithmetik der Polynome. Math. Z. 19, 247–264 (1924)
Kanold, H.-J.: Über eine zahlentheoretische Funktion von Jacobsthal. Math. Ann. 170, 314–326 (1967)
Kantor, J.-M.: Sur le polynôme associé à un polytope à sommets entiers dans R n. C. R. Acad. Sci. Paris 314, 669–672 (1992)
Kantor, J.-M., Khovanskii, A.: Une application du théorème de Riemann-Roch combinatoire au polynôme d’Ehrhart des polytopes entiers de R d. C. R. Acad. Sci. Paris 317, 501–507 (1993)
Kaplansky, I.: Ternary positive quadratic forms that represent all odd positive integers. Acta Arith. 70, 209–214 (1995)
Karatsuba, A.A., Kovalevskaya, E.I., Kubilyus, I.P. [Kubilius, J.]: Biography and scientific work of V.G. Sprindzhuk. Acta Arith. 53, 5–16 (1989) (in Russian)
Kasch, F., Volkmann, B.: Zur Mahlerschen Vermutung über S-Zahlen. Math. Ann. 136, 442–453
Katz, N.M., Tate, J.: Bernard Dwork (1923–1998). Not. Am. Math. Soc. 46, 338–343 (1999)
Kawada, K., Wooley, T.D.: Sums of fifth powers and related topics. Acta Arith. 87, 27–65 (1998)
Keates, M.: On the greatest prime factor of a polynomial. Proc. Edinb. Math. Soc. 16, 301–303 (1968/1969)
Keating, J.P., Snaith, N.C.: Random matrix theory and \(\zeta(\frac{1}{2}+it)\). Commun. Math. Phys. 214, 57–89 (2000)
Kempner, A.: On transcendental numbers. Trans. Am. Math. Soc. 17, 476–482 (1916)
Kersten, I.: Ernst Witt 1911–1991. Jahresber. Dtsch. Math.-Ver. 95, 166–180 (1993)
Kesseböhmer, M., Zhu, S.: Dimension sets for infinite IFSs: the Texan conjecture. J. Number Theory 116, 230–246 (2006)
Khintchine, A.J.: Ein Satz über Kettenbrüche, mit arithmetischen Anwendungen. Math. Z. 18, 289–306 (1923)
Khintchine, A.J.: Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math. Ann. 92, 115–125 (1924)
Khintchine, A.J.: Zur metrischen Theorie der diophantischen Approximationen. Math. Z. 24, 706–714 (1926)
Kifer, Y., Peres, Y., Weiss, B.: A dimension gap for continued fractions with independent digits. Isr. J. Math. 124, 61–76 (2001)
Kiming, I., Wang, X.D.: Examples of 2-dimensional, odd Galois representations of A 5-type over Q satisfying the Artin conjecture. In: Lecture Notes in Math., vol. 1585, pp. 109–121. Springer, Berlin (1994)
Kinney, J.R., Pitcher, T.S.: The dimension of some sets defined in terms of f-expansions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 4, 293–315 (1965/1966)
Kirmse, J.: Zur Darstellung total positiver Zahlen als Summen von vier Quadraten. Math. Z. 21, 195–202 (1924)
Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge University Press, Cambridge (1993)
Klein, F.: Ueber die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades. Math. Ann. 14, 111–172 (1879)
Kloosterman, H.D.: Over het splitsen van geheele postieve getallen in een som van kwadraten. Dissertation, Univ. of Groningen (1923)
Kloosterman, H.D.: On the representation of numbers in the form ax 2+by 2+cz 2+dt 2. Proc. Lond. Math. Soc. 25, 143–173 (1926)
Kloosterman, H.D.: On the representation of numbers in the form ax 2+by 2+cz 2+dt 2. Acta Math. 49, 407–464 (1926)
Kloosterman, H.D.: Thetareihen in total-reellen algebraischen Zahlkörpern. Math. Ann. 103, 279–299 (1930)
Kneser, M.: Zum expliziten Reziprozitätsgesetz von I.R. Shafarevič. Math. Nachr. 6, 89–96 (1951)
Kneser, M.: Martin Eichler (1912–1992). Acta Arith. 65, 293–296 (1993)
Koch, H.: Nachruf auf Hans Reichardt. Jahresber. Dtsch. Math.-Ver. 95, 135–140 (1993)
von Koch, H.: Sur la distribution des nombres premiers. Acta Math. 24, 159–182 (1901)
von Koch, H.: Ueber die Riemannsche Primzahlfunction. Math. Ann. 55, 440–464 (1902)
Koksma, J.F.: Diophantische Approximationen. Springer, Berlin (1936) [Reprint: Springer, 1974]
Kolesnik, G.: On the order of \(\zeta(\frac{1}{2}+it)\) and Δ(R). Pac. J. Math. 98, 107–122 (1982)
Kolesnik, G.: An improvement of the method of exponent pairs. In: Topics in Classical Number Theory. Colloq. Math. Soc. J. Bólyai, vol. 34, pp. 907–926. North-Holland, Amsterdam (1984)
Kolesnik, G.: On the method of exponent pairs. Acta Arith. 45, 115–143 (1985)
Korobov, N.M.: On zeros of the function ζ(s). Dokl. Akad. Nauk SSSR 118, 431–432 (1958) (in Russian)
Korobov, N.M.: Weyl’s estimates of sums and the distribution of primes. Dokl. Akad. Nauk SSSR 123, 28–31 (1958) (in Russian)
Kotov, S.V.: On the largest prime divisor of a polynomial. Mat. Zametki 13, 515–522 (1973) (in Russian)
Kowalski, E.: Some local-global applications of Kummer theory. Manuscr. Math. 111, 105–139 (2003)
Kozlov, V.Ya., Mardzanišvili, K.K.: Andreĭ Borisovič Šidlovskiĭ (on the occasion of his sixtieth birthday). Usp. Mat. Nauk 31(3), 225–232 (1976) (in Russian)
Kronecker, L.: Über die Irreductibilität der Gleichungen. Mon. Ber. Kgl. Preuß. Akad. Wiss., 1880, 155–163 [[3532], vol. 2, pp. 85–93]
Kronecker, L.: Grundzüge einer arithmetischen Theorie der algebraischen Grössen. J. Reine Angew. Math. 92, 1–122 (1882) [[3532], vol. 2, pp. 237–387]
Krull, W.: Galoissche Theorie der unendlichen algebraischen Erweiterungen. Math. Ann. 100, 687–698 (1928)
Krull, W.: Idealtheorie in unendlichen Zahlkörpern. Math. Z. 29, 42–54 (1928)
Krull, W.: Idealtheorie in unendlichen Zahlkörpern, II. Math. Z. 31, 527–557 (1930)
Kulas, M.: Refinement of an estimate for the Hurwitz zeta function in a neighbourhood of the line σ=1. Acta Arith. 89, 301–309 (1999)
Kummer, E.E.: Über die allgemeinen Reziprozitätsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist. Abhandl. Kgl. Preuss. Akad. Wiss. Berlin, 1859, 19–159. [[3583], vol. 1, pp. 699–839]
Kuzel, A.V.: Elementary solution of Waring’s problem for polynomials using Yu.V. Linnik’s method. Usp. Mat. Nauk 11(3), 165–168 (1956) (in Russian)
Kuzmin, R.O.: On a problem of Gauss. Dokl. Akad. Nauk SSSR, 1928, 375–380 (in Russian)
Kuzmin, R.O.: Sur un problème de Gauss. In: Atti congresso int. Mat., Bologna, 1928, vol. 6, pp. 83–89 (1932)
Lagarias, J.C.: Sets of primes determined by systems of polynomial congruences. Ill. J. Math. 27, 224–239 (1983)
Lagarias, J.C., Montgomery, H.L., Odlyzko, A.M.: A bound for the least ideal in the Chebotarev density theorem. Invent. Math. 54, 271–296 (1979)
Landau, E.: Ueber die zahlentheoretische Funktion ϕ(n) und ihre Beziehung zum Goldbachschen Satz. Nachr. Ges. Wiss. Göttingen, 1900, 177–186. [[3680], vol. 1, pp. 106–115]
Landau, E.: Über die Zerlegung total positiver Zahlen in Quadrate. Nachr. Ges. Wiss. Göttingen, 1919, 392–396 [[3680], vol. 7, pp. 201–204]
Landau, E.: Zur Hardy-Littlewoodschen Lösung des Waringschen Problems. Nachr. Ges. Wiss. Göttingen, 1921, 88–92 [[3680], vol. 7, pp. 327–331]
Landau, E.: Zum Waringschen Problem. Math. Z. 12, 219–247 (1922) [[3680], vol. 7, pp. 383–411]
Landau, E.: Zur additiven Primzahltheorie. Rend. Circ. Mat. Palermo 46, 349–356 (1922) [[3680], vol. 7, pp. 436–443]
Landau, E.: Über die ζ-Funktion und die L-Funktionen. Math. Z. 20, 105–125 (1924) [[3680], vol. 8, pp. 77–97]
Landau, E.: Die Winogradowsche Methode zum Beweise des Waring-Hilbert-Kamkeschen Satzes. Acta Math. 48, 217–253 (1926) [[3680], vol. 8, pp. 307–343]
Landau, E.: Vorlesungen über Zahlentheorie, vols. I–III. Hirzel, Leipzig (1927) [Reprint: Chelsea, 1950, 1969; English translation of vol. I: Elementary Number Theory, Chelsea, 1958]
Landau, E.: Über die neue Winogradoffsche Behandlung des Waringschen Problems. Math. Z. 31, 319–338 (1929) [[3680], vol. 9, pp. 137–156]
Lander, L.J., Parkin, T.R.: On first appearance of prime differences. Math. Comput. 21, 483–488 (1967)
Lang, S.: Integral points on curves. Publ. Math. Inst. Hautes Études Sci. 6, 27–43 (1960)
Lang, S.: A transcendence measure for E-functions. Mathematika 9, 157–161 (1962)
Lang, S.: Algebraic Number Theory. Addison-Wesley, Reading (1970) [2nd ed. Springer, 1994]
Lang, S., Néron, A.: Rational points of abelian varieties over function fields. Am. J. Math. 81, 95–118 (1959)
Langlands, R.P.: On Artin’s L-functions. In: Complex Analysis, Rice University Studies, vol. 56, pp. 23–28. Rice University, Houston (1970)
Langlands, R.P.: Problems in the theory of automorphic forms. In: Lecture Notes in Math., vol. 170, pp. 18–61. Springer, Berlin (1970)
Lavrik, A.F.: Approximate functional equations of Dirichlet functions. Izv. Akad. Nauk SSSR, Ser. Mat. 32, 134–185 (1968) (in Russian)
Leep, D.B., Schueller, L.M.: Classification of pairs of symmetric and alternating bilinear forms. Expo. Math. 17, 385–414 (1999)
Legendre, A.M.: Essai sur la théorie des nombres, Duprat, Paris (1798) [2nd ed. 1808, 3rd ed.: Théorie des nombres, Paris, 1830; German translation: Teubner, 1886, 1894]
Lehman, R.S.: Separation of zeros of the Riemann zeta-function. Math. Comput. 20, 523–541 (1966)
Lehmer, D.H.: The lattice points of an n-dimensional tetrahedron. Duke Math. J. 7, 341–353 (1940)
Lehmer, D.H.: On the roots of the Riemann zeta-function. Acta Math. 95, 291–298 (1956)
Lehmer, D.H.: Extended computation of the Riemann zeta-function. Mathematika 3, 102–108 (1956)
Lehmer, D.H., Lehmer, E.: On runs of residues. Proc. Am. Math. Soc. 13, 102–106 (1962)
Lehmer, D.H., Lehmer, E., Mills, W.H.: Pairs of consecutive power residues. Can. J. Math. 15, 172–177 (1963)
Lehmer, D.H., Lehmer, E., Mills, W.H., Selfridge, J.L.: Machine proof of a theorem on cubic residues. Math. Comput. 16, 407–415 (1962)
Lemmermeyer, F.: A note on Pépin’s counter-examples to Hasse principle for curves of genus 1. Abh. Math. Semin. Univ. Hamb. 69, 335–345 (1999)
Lemmermeyer, F.: Reciprocity Laws. Springer, Berlin (2000)
Lenskoĭ, D.N., Linnik, Yu.V.: Nikolai Grigorevič Čudakov (on his sixtieth birthday). Usp. Mat. Nauk 20(2), 237–240 (1965) (in Russian)
Lenz, H., et al.: Richard Rado, 1906–1989. Jahresber. Dtsch. Math.-Ver. 93, 127–145 (1991)
LeVeque, W.J.: Topics in Number Theory. Addison-Wesley, Reading (1956)
Levinson, N.: Ω-theorems for the Riemann zeta-function. Acta Arith. 20, 317–330 (1972)
Lévy, P.: Sur les lois de probabilité dont dépend les quotients complets et incomplets d’une fraction continue. Bull. Soc. Math. Fr. 57, 178–194 (1929)
Lin, K.-P., Yau, S.S.-T.: A sharp upper estimate of the number of integral points in a 5-dimensional tetrahedra. J. Number Theory 93, 207–234 (2002)
Lindemann, F.: Über die Zahl π. Math. Ann. 20, 213–225 (1882)
Lindemann, F.: Über den Fermatschen Satz betreffend die Unmöglichkeit der Gleichung x n=y n+z n. Sitzungsber. - Bayer. Akad. Wiss., Philos.- Hist. Kl. 31, 185–202 (1901)
Lindemann, F.: Über das sogenannte letzte Fermatsche Theorem. Sitzungsber. - Bayer. Akad. Wiss., Philos.- Hist. Kl. 37, 287–305 (1907)
Lindemann, F.: Über den sogenannten letzten Fermatschen Satz. Teubner, Leipzig (1909)
Linnik, Yu.V.: A new proof of the Goldbach–Vinogradov theorem. Mat. Sb. 19, 3–8 (1946) (in Russian) [[3929], vol. 2, pp. 23–27]
Linnik, Yu.V.: Asymptotical distribution of integral points on a sphere. Dokl. Akad. Nauk SSSR 96, 909–912 (1954) (in Russian) [[3929], vol. 2, pp. 134–138]
Linnik, Yu.V.: Asymptotical-geometric and ergodic properties of the set of integral points on a sphere. Mat. Sb. 43, 257–276 (1957) (in Russian) [[3929], vol. 2, pp. 209–228]
Linnik, Yu.V.: All large numbers are sums of a prime and two squares (On a problem of Hardy and Littlewood), I. Mat. Sb. 52, 661–700 (1960) (in Russian) [[3929], vol. 2, pp. 217–289]
Linnik, Yu.V.: All large numbers are sums of a prime and two squares (On a problem of Hardy and Littlewood), II. Mat. Sb. 53, 3–38 (1961) (in Russian) [[3929], vol. 2, pp. 217–289]
Linnik, Yu.V.: An asymptotic formula in an additive problem of Hardy-Littlewood. Izv. Akad. Nauk SSSR, Ser. Mat. 24, 629–706 (1960) (in Russian)
Linnik, Yu.V.: Ergodic Properties of Algebraic Fields. Izdat. Leningrad. Univ., Leningrad (1967) (in Russian) [English translation: Springer, 1968]
Linnik, Yu.V., Malyšev, A.V.: On integral points on a sphere. Dokl. Akad. Nauk SSSR 89, 209–211 (1953) (in Russian) [[3929], vol. 1, pp. 128–133]
Liouville, J.: Sur de classes très eténdues de quantités dont la valeur n’est ni algébrique, ni même reductible à des irrationelles algébriques. C. R. Acad. Sci. Paris 18, 883–885 (1844)
Liouville, J.: Sur de classes très eténdues de quantités dont la valeur n’est ni algébrique, ni même reductible à des irrationelles algébriques, II. J. Math. Pures Appl. 16, 133–142 (1851)
Littlewood, J.E.: Researches in the theory of the Riemann ζ-function. Proc. Lond. Math. Soc. 20, xxii–xxviii (1922)
Littlewood, J.E.: On the Riemann zeta-function. Proc. Lond. Math. Soc. 24, 175–201 (1925)
Liu, H.Q.: On a fundamental result in van der Corput’s method of estimating exponential sums. Acta Arith. 90, 357–370 (1999)
Liu, M.C.: On a theorem of Heilbronn concerning the fractional part of θn 2. Can. J. Math. 22, 784–788 (1970)
Liu, M.C.: Simultaneous approximation of additive forms. Trans. Am. Math. Soc. 206, 361–373 (1975)
Ljunggren, W.: A note on simultaneous Pell equations. Norsk Mat. Tidsskr. 23, 132–138 (1941) (in Norwegian)
Lochs, G.: Über die Lösungszahl einer linearen, diophantischen Ungleichung. Jahresber. Dtsch. Math.-Ver. 54, 41–51 (1950)
Lochs, G.: Über die Anzahl der Gitterpunkte in einem Tetraeder. Monatshefte Math. 56, 233–239 (1952)
Lomadze, G.A.: On the representation of numbers by sums of squares. Tr. Tbil. Inst. Razmadze 16, 231–275 (1948) (in Russian)
Lomadze, G.A.: On the representation of numbers by sums of squares. Tr. Tbil. Inst. Razmadze 20, 47–87 (1954) (in Russian)
London, H., Finkelstein, R.: On Mordell’s Equation y 2−k=x 3. Bowling Green State University Press, Bowling Green (1973)
Lou, S.T., Yao, Q.: A Chebychev’s type of prime number theorem in a short interval, II. Hardy-Ramanujan J. 15, 1–33 (1992)
Lou, S.T., Yao, Q.: The number of primes in a short interval. Hardy-Ramanujan J. 16, 21–43 (1993)
Luca, F., Mukhopadhyay, A., Srinivas, K.: Some results on Oppenheim’s “Factorisatio Numerorum” function. Acta Arith. 142, 41–50 (2010)
Maass, H.: Über die Darstellung total positiver Zahlen des Körpers \(R(\sqrt{5})\) als Summe von drei Quadraten. Abh. Math. Semin. Hansischen Univ. 14, 185–191 (1941)
Macdonald, I.G.: The volume of a lattice polyhedron. Proc. Camb. Philos. Soc. 59, 719–726 (1963)
Macdonald, I.G.: Polynomials associated with finite cell-complexes. J. Lond. Math. Soc. 4, 181–192 (1971)
Maclaurin, C.: Treatise of Fluxions. Ruddimans, Edinburgh (1742)
MacMahon, P.A.: Dirichlet series and the theory of partitions. Proc. Lond. Math. Soc. 22, 404–411 (1923)
Magnus, W.: Über den Beweis des Hauptidealsatzes. J. Reine Angew. Math. 170, 235–240 (1934)
Mahler, K.: Zur Approximation der Exponentialfunktion und des Logarithmus, I. J. Reine Angew. Math. 166, 118–136 (1932)
Mahler, K.: Zur Approximation der Exponentialfunktion und des Logarithmus, II. J. Reine Angew. Math. 166, 137–150 (1932)
Mahler, K.: Über den größten Primteiler der Polynome x 2±1. Arch. Math. Naturvidensk. 41(1), 1–8 (1933)
Mahler, K.: Über rationalen Punkte auf Kurven vom Geschlecht Eins. J. Reine Angew. Math. 170, 168–178 (1934)
Mahler, K.: Über den größten Primteiler spezieller Polynomen zweiten Grades. Arch. Math. Naturvidensk. 41(6), 1–26 (1935)
Mahler, K.: Note on hypothesis K of Hardy and Littlewood. J. Lond. Math. Soc. 11, 136–138 (1936)
Mahler, K.: On the approximation of π. Indag. Math. 15, 30–42 (1953)
Mahler, K.: Lectures on Transcendental Numbers. Lecture Notes in Math., vol. 546. Springer, Berlin (1976)
Maier, H.: Primes in short intervals. Mich. Math. J. 32, 221–225 (1985)
Maier, H., Pomerance, C.: Unusually large gaps between consecutive primes. Trans. Am. Math. Soc. 322, 201–237 (1990)
Maillet, E.: Quelques extensions du théorème de Fermat sur les nombres polygones. J. Math. Pures Appl. 2, 363–380 (1896)
Maillet, E.: Introduction à la théorie des nombres transcendants et des propriétes arithmétiques des fonctions. Gauthier-Villars, Paris (1906)
Maillet, E.: Détermination des points entiers des courbes algébriques unicursales à coefficients entiers. C. R. Acad. Sci. Paris 168, 217–220 (1919)
Maillet, E.: Détermination des points entiers des courbes algébriques unicursales à coefficients entiers dans l’espace à k dimensions. J. Éc. Polytech. 20, 115–156 (1920)
Malyšev, A.V.: On the representation of integers by positive quadratic forms. Tr. Mat. Inst. Steklova 65, 1–212 (1962) (in Russian)
Manin, Yu.I.: Le groupe de Brauer-Grothendieck en géométrie diophantienne. Actes ICM Nice 1, 401–411 (1970)
Manin, Yu.I.: Cubic Forms. Nauka, Moscow (1972) (in Russian) [English translation: North-Holland, 1974, 2nd ed. 1986]
Mardžanišvili, K.K.: On the simultaneous representations of n integers by sums of complete first, second, n-th powers. Izv. Akad. Nauk SSSR, Ser. Mat. 1, 609–631 (1937) (in Russian)
Mardžanišvili, K.K.: Sur un système d’équations de Diophante. Dokl. Akad. Nauk SSSR 22, 467–470 (1939)
Mardžanišvili, K.K.: On an additive problem in number theory. Izv. Akad. Nauk SSSR, Ser. Mat. 4, 193–213 (1940) (in Russian)
Mardžanišvili, K.K.: On an asymptotic formula of the additive theory of prime numbers. Soobšč. AN Gruz. SSR 8, 597–604 (1997) (in Russian)
Margulis, G.A.: Formes quadratiques indéfinies et flots unipotents sur les espaces homogènes. C. R. Acad. Sci. Paris 304, 249–253 (1987)
Margulis, G.A.: Discrete subgroups and ergodic theory. In: Number Theory, Trace Formulas and Discrete Groups, Oslo, 1989, pp. 377–398 (1989)
Margulis, G.A.: Indefinite quadratic forms and unipotent flows on homogeneous spaces. Banach Cent. Publ. 23, 399–409 (1989)
Marke, P.W.: Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung. Math. Ann. 114, 29–56 (1937)
Markoff, A.A.: Dèmonstration d’un théorème de Tchébycheff. C. R. Acad. Sci. Paris 120, 1032–1034 (1895)
Marstrand, J.M.: On Khinchin’s conjecture about strong uniform distribution. Proc. Lond. Math. Soc. 21, 540–556 (1970)
Masser, D.W.: A note on Siegel’s lemma. Rocky Mt. J. Math. 26, 1057–1068 (1996)
Masser, D.W., Rickert, J.H.: Simultaneous Pell equations. J. Number Theory 61, 52–66 (1996)
Maxwell, J.W.: William J. LeVeque (1923–2007). Not. Am. Math. Soc. 55, 1261–1262 (2008)
Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18, 183–226 (1972)
Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. Inst. Hautes Études Sci. 47, 33–186 (1977)
Mazur, B.: On the passage from local to global in number theory. Bull. Am. Math. Soc. 29, 14–50 (1993)
Meyer, A.: Ueber die Kriterien für die Auflösbarkeit der Gleichung ax 2+by 2+cz 2+du 2=0 in ganzen Zahlen. Vierteljahr. Naturforsch. Ges. Zürich 29, 209–222 (1884)
Min, S.-H.: On the order of ζ(1/2+it). Trans. Am. Math. Soc. 65, 448–472 (1949)
Minkowski, H.: Zur Theorie der positiven quadratischen Formen. J. Reine Angew. Math. 101, 196–202 (1887) [[4329], vol. 1, pp. 212–239]
Minkowski, H.: Über die Bedingungen, unter welchen zwei quadratische Formen mit rationalen Koeffizienten ineinander transformiert werden können. J. Reine Angew. Math. 106, 5–26 (1890) [[4329], vol. 1, pp. 219–239]
Mitkin, D.A.: Estimate for the number of summands in the Hilbert-Kamke problem. Mat. Sb. 129, 549–577 (1986) (in Russian)
Mitkin, D.A.: Estimate for the number of summands in the Hilbert-Kamke problem, II. Mat. Sb. 132, 345–351 (1987) (in Russian)
Mitkin, D.A.: The Hilbert-Kamke problem in prime numbers. Usp. Mat. Nauk 42(5), 205–206 (1987) (in Russian)
Montgomery, H.L.: Zeros of L-functions. Invent. Math. 8, 346–354 (1969)
Montgomery, H.L., Vaughan, R.C.: The large sieve. Mathematika 20, 119–134 (1973)
Mordell, L.J.: The diophantine equation y 2−k=x 3. Proc. Lond. Math. Soc. 13, 60–80 (1913)
Mordell, L.J.: A statement by Fermat. Proc. Lond. Math. Soc. 18, v (1919)
Mordell, L.J.: On the rational solutions of the indeterminate equations of the third and fourth degrees. Proc. Camb. Philos. Soc. 21, 179–192 (1922)
Mordell, L.J.: A Chapter in the Theory of Numbers. Cambridge University Press, Cambridge (1947)
Mordell, L.J.: On some Diophantine equations y 2=x 3+k with no rational solutions. Arch. Math. Naturvidensk. 49, 143–150 (1947)
Mordell, L.J.: On some Diophantine equations y 2=x 3+k with no rational solutions, II. In: 1969 Number Theory and Analysis, pp. 223–232. Plenum, New York (1969)
Mordell, L.J.: Rational points on cubic surfaces. Publ. Math. (Debr.) 1, 1–6 (1949)
Mordell, L.J.: Lattice points in a tetrahedron and generalized Dedekind sums. J. Indian Math. Soc. 15, 41–46 (1951)
Mordell, L.J.: On the conjecture for the rational points on a cubic surface. J. Lond. Math. Soc. 40, 149–158 (1965)
Mordell, L.J.: The infinity of rational solutions of y 2=x 3+k. J. Lond. Math. Soc. 41, 523–525 (1966)
Mordell, L.J.: On the magnitude of the integer solutions of the equation ax 2+by 2+cz 2=0. J. Number Theory 1, 1–3 (1969)
Moree, P.: Approximation of singular series and automata. Manuscr. Math. 101, 385–399 (2000)
Morelli, R.: Pick’s theorem and the Todd class of a toric variety. Adv. Math. 100, 183–231 (1993)
Moreno, C.J., Wagstaff, S.S. Jr.: Sums of Squares of Integers. Chapman & Hall/CRC, London/Boca Raton (2006)
Moriya, M.: Rein arithmetisch-algebraischer Aufbau der Klassenkörpertheorie über algebraischen Funktionenkörpern einer Unbestimmten mit endlichem Konstantenkörper. Jpn. J. Math. 14, 67–84 (1938)
Moser, L.: Notes on number theory. II. On a theorem of van der Waerden. Can. Math. Bull. 3, 23–25 (1960)
Moshchevitin, N.G.: On small fractional parts of polynomials. J. Number Theory 129, 349–357 (2009)
Mostow, G.D.: Abraham Robinson, 1918–1974. Isr. J. Math. 25, 4–14 (1976)
Mozzochi, C.J.: On the difference between consecutive primes. J. Number Theory 24, 181–187 (1986)
Mueller, J.: On the difference between consecutive primes. In: Recent Progress in Analytic Number Theory, vol. 1, pp. 269–273. Academic Press, San Diego (1981)
Nagel[l], T.: Généralisation d’un théorème de Tchebycheff. J. Math. Pures Appl. 4, 343–356 (1921)
Nagel[l], T.: Zur Arithmetik der Polynome. Abh. Math. Semin. Univ. Hamb. 1, 179–194 (1922)
Nagell, T.: Über Einheiten in reinen kubischen Körpern. Christiania Vid. Selsk, Skr., 1923, nr. 11, 1–34
Nagell, T.: Über die rationalen Punkte auf einigen kubischen Kurven. Tohoku Math. J. 24, 48–53 (1924/1925)
Nagell, T.: Über einige kubische Gleichungen mit zwei Unbestimmten. Math. Z. 24, 422–447 (1925)
Nagell, T.: Solution complète de quelques équations cubiques à deux indéterminées. J. Math. Pures Appl. 4, 209–270 (1925)
Nagell, T.: Darstellung ganzer Zahlen durch binäre kubische Formen mit negativer Diskriminante. Math. Z. 28, 10–29 (1928)
Nagell, T.: Sur les propriétés arithmétiques des cubiques planes du premier genre. Acta Math. 52, 93–126 (1928)
Nagell, T.: Über den grössten Primteiler gewisser Polynome dritten Grades. Math. Ann. 114, 284–292 (1937)
Nair, M.: Power free values of polynomials. Mathematika 23, 159–183 (1976)
Nair, M.: Power free values of polynomials, II. Proc. Lond. Math. Soc. 38, 353–368 (1979)
Nair, R.: On strong uniform distribution. Acta Arith. 56, 183–193 (1990)
Nair, R.: On strong uniform distribution, II. Monatshefte Math. 132, 341–348 (2001)
Nair, R.: On strong uniform distribution, III. Indag. Math. 14, 233–240 (2003)
Nair, R.: On strong uniform distribution, IV. J. Inequal. Appl., 2005, 319–327
Nakagawa, J., Horie, K.: Elliptic curves with no rational points. Proc. Am. Math. Soc. 104, 20–24 (1988)
Narkiewicz, W.: The Development of Prime Number Theory. Springer, Berlin (2000)
Nečaev, V.I.: Waring’s problem for polynomials. Tr. Mat. Inst. Steklova 38, 190–243 (1951) (in Russian)
Néron, A.: Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps. Bull. Soc. Math. Fr. 80, 101–166 (1952)
Nesterenko, Yu.V.: Estimate of the orders of the zeroes of functions of a certain class, and their application in the theory of transcendental numbers. Izv. Akad. Nauk SSSR, Ser. Mat. 41, 253–284 (1977) (in Russian)
Nesterenko, Yu.V.: On the number π. Vestnik Moskov. Univ. Ser. I. Mat. Mekh., 1987, nr. 3, 7–10 (in Russian)
Neukirch, J.: Class Field Theory. Springer, Berlin (1986)
Nicely, T.R.: New maximal prime gaps and first occurrences. Math. Comput. 68, 1311–1315 (1999)
Nikolskiĭ, S.M., Zaharov, V.K.: Konstantin Konstantinovič Mardžanišvili. Usp. Mat. Nauk 38(5), 107–109 (1983) (in Russian)
Norton, K.K.: Numbers with small prime factors, and the least kth power non-residue. Mem. Am. Math. Soc. 106, 1–106 (1971)
Oesterlé, J.: Effective versions of the Chebotarev density theorem. Astérisque 61, 165–167 (1979)
Okano, T.: A note on the rational approximations to e. Tokyo J. Math. 15, 129–133 (1992)
Okazaki, R.: Geometry of a cubic Thue equation. Publ. Math. (Debr.) 61, 267–314 (2002)
Oppenheim, A.: On an arithmetic function. J. Lond. Math. Soc. 1, 205–211 (1926)
Oppenheim, A.: On an arithmetic function, II. J. Lond. Math. Soc. 2, 123–130 (1927)
Oppenheim, A.: The minima of indefinite quaternary quadratic forms. Proc. Natl. Acad. Sci. USA 15, 724–727 (1929)
Oppenheim, A.: The determination of all universal ternary quadratic forms. Q. J. Math. 1, 179–185 (1930)
Oppenheim, A.: Values of quadratic forms, I. Q. J. Math. 4, 54–59 (1953)
Oppenheim, A.: Values of quadratic forms, II. Q. J. Math. 4, 60–66 (1953)
Ostrowski, A.: Bemerkungen zur Theorie der Diophantischen Approximationen. Abh. Math. Semin. Univ. Hamb. 1, 77–98 (1922); corr.: 250–251; 4, 224 (1922)
Pall, G.: Large positive integers are sums of four or five values of a quadratic function. Am. J. Math. 54, 66–78 (1932)
Pall, G.: An almost universal form. Bull. Am. Math. Soc. 46, 291 (1940)
Pall, G.: The completion of a problem of Kloosterman. Am. J. Math. 68, 47–58 (1946)
Panaitopol, L.: A special case of the Hardy-Littlewood conjecture. Math. Rep. (Bucuresti) 4, 265–268 (2002)
Pépin, T.: Sur certains nombres complexes compris dans la formule \(a + b\sqrt{-c}\). J. Math. Pures Appl. 1, 317–372 (1875)
Pépin, T.: Théorèmes d’analyse indéterminée. C. R. Acad. Sci. Paris 88, 1255–1257 (1879)
Pépin, T.: Démonstration du théorème de Fermat sur les nombres polygones. Atti Accad. Naz. Lincei 46, 119–131 (1893)
Peters, M.: Darstellungen durch definite ternäre quadratische Formen. Acta Arith. 34, 57–80 (1977/1978)
Petrov, F.V.: On the number of rational points on a strictly convex curve. Funkc. Anal. Prilozh. 40, 30–42 (2006) (in Russian)
Peyre, E.: Obstructions au principe de Hasse et à l’approximation faible. Astérisque 299, 165–193 (2005)
Pezda, T.: Cycles of polynomial mappings in several variables over rings of integers in finite extensions of the rationals. Acta Arith. 108, 127–146 (2003)
Phillips, E.: The zeta-function of Riemann: Further developments of van der Corput’s method. Q. J. Math. 4, 209–225 (1933)
Pick, G.: Ueber gewisse ganzzahlige lineare Substitutionen, welche sich nicht durch algebraische Congruenzen erklären lassen. Math. Ann. 28, 119–124 (1886)
Pila, J.: Geometric postulation of a smooth function and the number of rational points. Duke Math. J. 63, 449–463 (1991)
Pila, J.: Density of integer points on plane algebraic curves. Internat. Math. Res. Notices, 1996, 903–912
Piltz, A.: Über die Häufigkeit der Primzahlen in arithmetischen Progressionen und über verwandte Gesetze. Habilitationsschrift, Jena (1884)
Pintz, J.: On primes in short intervals, I. Studia Sci. Math. Hung. 16, 395–414 (1981)
Pintz, J.: On primes in short intervals, II. Studia Sci. Math. Hung. 19, 89–96 (1984)
Pintz, J.: Very large gaps between consecutive primes. J. Number Theory 63, 286–301 (1997)
Pintz, J.: Recent results on the Goldbach problem. In: Elementary and Analytic Number Theory. Proceedings of the ELAZ Conference, Stuttgart, May 24–28, 2004, pp. 220–254 (2006)
Plagne, A.: A uniform version of Jarník’s theorem. Acta Arith. 87, 255–267 (1999)
Podsypanin, V.G.: On the indeterminate equation x 3=y 2+Az 6. Mat. Sb. 24, 391–403 (1949) (in Russian)
Poincaré, H.: Sur les propriétés arithmétiques des courbes algébriques. J. Math. Pures Appl. 7, 161–233 (1901)
Poisson, S.D.: Mémoire sur le calcul numérique des intégrales définies. Mém. de l’Institut 6, 571–602 (1823)
Pollington, A.D., Vaughan, R.C.: The k-dimensional Duffin and Schaeffer conjecture. J. Théor. Nr. Bordx. 1, 81–88 (1989)
Pollington, A.D., Vaughan, R.C.: The k-dimensional Duffin and Schaeffer conjecture. Mathematika 37, 190–200 (1990)
Pólya, G.: Généralisation d’un théorème de M. Störmer. Arch. Math. Naturvidensk. 35(5), 1–8 (1917) (in Swedish)
Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5, 227–257 (1959); corr., 7, 279 (1961/1962)
Pommersheim, J.E.: Toric varieties, lattice points and Dedekind sums. Math. Ann. 295, 1–24 (1993)
Poonen, B.: An explicit algebraic family of genus-one curves violating the Hasse principle. J. Théor. Nr. Bordx. 13, 263–274 (2001)
Popken, J.: Zur Transzendenz von e. Math. Z. 29, 525–541 (1929)
Popken, J.: Zur Transzendenz von π. Math. Z. 29, 542–548 (1929)
Potter, H.S.A.: The mean values of certain Dirichlet series, I. Proc. Lond. Math. Soc. 46, 467–478 (1940)
Potter, H.S.A.: The mean values of certain Dirichlet series, II. Proc. Lond. Math. Soc. 47, 1–19 (1940)
Poulakis, D.: Bounds for the minimal solutions of genus zero diophantine equations. Acta Arith. 86, 51–90 (1998)
Prasad, D., Yogananda, C.S.: A report on Artin’s holomorphy conjecture. In: Bambah, R.P., et al. (ed.) Number Theory, pp. 301–314. Birkhäuser, Basel (2000)
Rademacher, H.: Zur additiven Primzahltheorie algebraischer Zahlkörper, I. Über die Darstellung totalpositiver Zahlen als Summe von totalpositiven Primzahlen im reell-quadratischen Zahlkörper. Abh. Math. Semin. Univ. Hamb. 3, 109–163 (1924) [[5040], vol. 1, pp. 306–360]
Rademacher, H.: Zur additiven Primzahltheorie algebraischer Zahlkörper, II. Über die Darstellung von Körperzahlen als Summe von Primzahlen im imaginär-quadratischen Zahlkörper. Abh. Math. Semin. Univ. Hamb. 3, 331–378 (1924) [[5040], vol. 1, pp. 362–409]
Rademacher, H.: Zur additiven Primzahltheorie algebraischer Zahlkörper, III. Über die Darstellung totalpositiver Zahlen als Summen von totalpositiven Primzahlen in einem beliebigen Zahlkörper. Math. Z. 27, 321–426 (1928) [[5040], vol. 1, pp. 445–550]
Rademacher, H.: Über eine Erweiterung des Goldbachschen Problems. Math. Z. 25, 627–657 (1926) [[5040], vol. 1, pp. 413–443]
Rado, R.: Verallgemeinerung eines Satzes von van der Waerden mit Anwendungen auf ein Problem der Zahlentheorie. SBer. Preuß. Akad. Wiss. Berlin, 1933, 589–596
Rados, G.: Über Kongruenzbedingung der rationalen Lösbarkeit von algebraischen Gleichungen. Math. Ann. 87, 78–83 (1922)
Ramachandra, K.: Application of a theorem of Montgomery and Vaughan to the zeta-function. J. Lond. Math. Soc. 10, 482–486 (1975)
Ramachandra, K.: Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series, I. Hardy-Ramanujan J. 1, 1–15 (1978)
Ramachandra, K.: Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series, II. Hardy-Ramanujan J. 3, 1–24 (1980)
Ramachandra, K.: On the Mean-Value and Omega-Theorems for the Riemann Zeta-function. Springer, Berlin (1995)
Ramachandra, K., Sankaranarayanan, A.: On some theorems of Littlewood and Selberg, I. J. Number Theory 44, 281–291 (1993)
Ramaré, O., Saouter, Y.: Short effective intervals containing primes. J. Number Theory 98, 10–33 (2003)
Ramaré, O., Schlage-Puchta, J.-C.: Improving on the Brun-Titchmarsh theorem. Acta Arith. 131, 351–366 (2008)
Ramaswami, V.: The number of positive integers ≤x and free of prime divisors >x c, and a problem of S.S. Pillai. Duke Math. J. 16, 99–109 (1949)
Ramharter, G.: Some metrical properties of continued fractions. Mathematika 30, 117–132 (1983)
Rane, V.V.: On an approximate functional equation for Dirichlet L-series. Math. Ann. 264, 137–145 (1983)
Rankin, R.A.: The difference between consecutive prime numbers, I. J. Lond. Math. Soc. 13, 242–247 (1938)
Rankin, R.A.: The difference between consecutive prime numbers, V. Proc. Edinb. Math. Soc. 13, 331–332 (1962/1963)
Rankin, R.A.: Van der Corput’s method and the theory of exponent pairs. Q. J. Math. 6, 147–153 (1955)
Redouaby, M.: Sur la méthode de van der Corput pour des sommes d’exponentielles. J. Théor. Nr. Bordx. 13, 583–607 (2001)
Redouaby, M., Sargos, P.: Sur la transformation B de van der Corput. Expo. Math. 17, 207–232 (1999)
Reichardt, H.: Über die Diophantische Gleichung ax 4+bx 2 y 2+cy 4=ez 2. Math. Ann. 117, 235–276 (1940/1941)
Reichardt, H.: Einige im Kleinem überall lösbare, im Großen unlösbare diophantische Gleichungen. J. Reine Angew. Math. 184, 12–18 (1942)
Rhin, G.: Sur les mesures d’irrationalité de certains nombres transcendants. Groupe d’étude en théorie analytique des nombres 1(exp. 28), 1–6 (1984–1985)
Ricci, G.: Ricerche aritmetiche sui polinomi. Rend. Circ. Mat. Palermo 57, 433–475 (1933)
Ricci, G.: Ricerche aritmetiche sui polinomi, II. Rend. Circ. Mat. Palermo 58, 190–208 (1934)
Ricci, G.: Su un teorema di Tchebychef-Nagel. Ann. Mat. Pura Appl. 12, 295–303 (1934)
Richert, H.-E.: On the difference between consecutive squarefree numbers. J. Lond. Math. Soc. 29, 16–20 (1954)
Richert, H.-E.: Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen σ=1. Math. Ann. 169, 97–101 (1967)
Ridout, D.: Indefinite quadratic forms. Mathematika 5, 122–124 (1958)
Rieger, G.J.: Die metrische Theorie der Kettenbrüche seit Gauß. Abh. Braunschw. Wiss. Ges. 27, 103–117 (1977)
Rignaux, M.: L’Intermédiaire Math. 25, 94–95 (1918)
Robinson, A., Roquette, P.: On the finiteness theorem of Siegel and Mahler concerning Diophantine equations. J. Number Theory 7, 121–176 (1975)
Rogers, C.A.: Richard Rado. Bull. Lond. Math. Soc. 30, 185–195 (1988)
Rohrbach, H.: Richard Brauer zum Gedächtnis. Jahresber. Dtsch. Math.-Ver. 83, 125–134 (1981)
Rohrbach, H., Weis, J.: Zum finiten Fall des Bertrandschen Postulats. J. Reine Angew. Math. 214/215, 432–440 (1964)
Rohrlich, D.E.: Elliptic curves with good reduction everywhere. J. Lond. Math. Soc. 25, 216–222 (1982)
Rohrlich, D.E.: On L-functions of elliptic curves and anticyclotomic towers. Invent. Math. 75, 383–408 (1984)
Romanoff, N.P.: Über einige Sätze der additiven Zahlentheorie. Math. Ann. 109, 668–678 (1934)
Roquette, P.: Artin’s Reciprocity Law and Hasse’s Local-Global Principle in Historical Perspective, to be published
Ross, A.E.: On criteria for universality of ternary quadratic forms. Q. J. Math. 4, 147–158 (1933)
Ross, A.E.: On a problem of Ramanujan. Am. J. Math. 68, 29–46 (1946)
Ross, A.E., Pall, G.: An extension of a problem of Kloosterman. Am. J. Math. 68, 59–65 (1946)
Rosser, J.B., Yohe, J.M., Schoenfeld, L.: Rigorous computation and the zeros of the Riemann zeta-function. In: Information Processing 68, vol. 1, pp. 70–76. North-Holland, Amsterdam (1969)
Roth, K.F.: On the gaps between squarefree numbers. J. Lond. Math. Soc. 26, 263–268 (1951)
Roth, K.F.: Rational approximations to algebraic numbers. Mathematika 2, 1–20 (1955); corr. p. 168
Roy, D., Thunder, J.L.: A note on Siegel’s lemma over number fields. Monatshefte Math. 120, 307–318 (1995)
Roy, D., Thunder, J.L.: An absolute Siegel’s lemma. J. Reine Angew. Math. 476, 1–26 (1996)
Rubin, K., Silverberg, A.: Ranks of elliptic curves. Bull. Am. Math. Soc. 39, 455–474 (2002)
Rubin, K., Wiles, A.: Mordell-Weil groups of elliptic curves over cyclotomic fields. In: Number Theory Related to Fermat’s Last Theorem, pp. 237–254. Birkhäuser, Basel (1982)
Rumely, R.S.: Arithmetic over the ring of all algebraic integers. J. Reine Angew. Math. 368, 127–133 (1986)
Ryll-Nardzewski, C.: On the ergodic theorem, II. Stud. Math. 12, 74–79 (1951)
Šafarevič, I.R.: A general reciprocity law. Mat. Sb. 26, 113–146 (1950) (in Russian)
Šafarevič, I.R.: Dmitriĭ Konstantinovič Faddeev. Algebra Anal. 2, 3–9 (1990) (in Russian)
Saias, E.: Sur le nombre des entiers sans grand facteur premier. J. Number Theory 32, 78–99 (1989)
Salberger, P.: On obstructions to the Hasse principle. In: Number Theory and Algebraic Geometry, pp. 251–277. Cambridge University Press, Cambridge (2003)
Salikhov, V.Kh.: Irreducibility of hypergeometric equations, and algebraic independence of values of E-functions. Acta Arith. 53, 453–471 (1990) (in Russian)
Salikhov, V.Kh.: On the irrationality measure of π. Usp. Mat. Nauk 63(3), 163–164 (2008) (in Russian)
Schinzel, A.: On two theorems of Gelfond and some of their applications. Acta Arith. 13, 177–236 (1967/1968)
Schinzel, A.: A refinement of a theorem of Gerst on power residues. Acta Arith. 17, 161–168 (1970)
Schinzel, A.: On power residues and exponential congruences. Acta Arith. 27, 397–420 (1975) [[5449], vol. 2, pp. 915–938]
Schinzel, A.: Hasse’s principle for systems of ternary quadratic forms and for one biquadratic form. Stud. Math. 77, 103–109 (1983) [[5449], vol. 1, pp. 87–92]
Schinzel, A., Sierpiński, W.: Sur certaines hypothèses concernant les nombres premiers. Acta Arith. 4, 185–208 (1958) [[5449], vol. 2, pp. 1113–1133]
Schlickewei, H.P.: S-unit equations over number fields. Invent. Math. 102, 95–107 (1990)
Schmidt, F.K.: Zur Klassenkörpertheorie im Kleinen. J. Reine Angew. Math. 162, 155–168 (1930)
Schmidt, F.K.: Die Theorie der Klassenkörper über einem Körper algebraischer Funktionen in einer Unbestimmten und mit endlichen Koeffizientenbereich. SBer. Erlangen 62, 267–284 (1931)
Schmidt, W.M.: Two combinatorial theorems on arithmetic progressions. Duke Math. J. 29, 129–140 (1962)
Schmidt, W.M.: Small Fractional Parts of Polynomials. Am. Math. Soc., Providence (1977)
Schmidt, W.M.: Integer points on curves and surfaces. Monatshefte Math. 99, 45–72 (1985)
Schmidt, W.M.: Integer points on hypersurfaces. Monatshefte Math. 102, 27–58 (1986)
Schneeberger, W.A.: Arithmetic and geometry of integral lattices. Ph.D. thesis, Princeton Univ. (1995)
Schoenfeld, L.: The order of the zeta function near the line σ=1. Duke Math. J. 24, 601–609 (1957)
Schoenfeld, L.: Sharper bounds for the Chebyshev functions θ(x) and ψ(x), II. Math. Comput. 30, 337–360 (1976); corr., 900
Scholz, A.: Die Abgrenzungssätze für Kreiskörper und Klassenkörper. SBer. Preuß. Akad. Wiss. Berl. 20, 417–426 (1931)
Scholz, A.: Totale Normenreste, die keine Normen sind, als Erzeuger nichtabelscher Körpererweiterungen, I. J. Reine Angew. Math. 175, 100–107 (1936)
Scholz, A.: Totale Normenreste, die keine Normen sind, als Erzeuger nichtabelscher Körpererweiterungen, II. J. Reine Angew. Math. 182, 217–234 (1940)
Scholz, A.: Zur Idealtheorie in unendlichen algebraischen Zahlkörpern. J. Reine Angew. Math. 185, 113–126 (1943)
Schönhage, A.: Eine Bemerkung zur Konstruktion grosser Primzahldifferenzen. Arch. Math. 14, 29–30 (1936)
Schreier, O.: Über eine Arbeit von Herrn Tschebotareff. Abh. Math. Semin. Univ. Hamb. 5, 1–6 (1927)
Schumann, H.G.: Zum Beweis des Hauptidealsatzes. Abh. Math. Semin. Univ. Hamb. 12, 42–47 (1937)
Schur, I.: Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, I. SBer. Preuß. Akad. Wiss. Berl. 23, 125–136 (1929) [[5578], vol. 3, pp. 140–151]
Schur, I.: Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, II. SBer. Preuß. Akad. Wiss. Berl. 23, 370–391 (1929) [[5578], vol. 3, 152–173]
Schwarz, W., Volkmann, B.: Hans Rohrbach zum Gedächtnis. 27.2.1903–19.12.1993. Jahresber. Dtsch. Math.-Ver. 105, 89–99 (2003)
Segal, S.L.: On π(x+y)≤π(x)+π(y). Trans. Am. Math. Soc. 104, 523–527 (1962)
Sekigawa, H., Koyama, K.: Nonexistence conditions of a solution for the congruence \(x^{k}_{1}+\cdots+x^{k}_{s}\equiv N\ (\mathrm{mod}\ p^{n})\). Math. Comput. 68, 1283–1297 (1999)
Selberg, A.: On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvidensk. 47, 87–105 (1943) [[5625], pp. 160–178]
Selberg, A.: The general sieve method and its place in prime number theory. In: Proc. ICM 1950, vol. 1, pp. 286–292. Am. Math. Soc., Providence (1952) [[5625], pp. 411–417]
Selberg, A.: Old and new conjectures and results about a class of Dirichlet series. In: Proceedings of the Amalfi Conference on Analytic Number Theory, Maiori, 1989, pp. 367–385. Univ. de Salerno, Salerno (1992). [[5626], pp. 47–63]
Selberg, A.: Lectures on sieves. In: Selberg, A. (ed.) Collected Papers, vol. 2, pp. 65–257. Springer, Berlin (1991)
Selmer, E.S.: The diophantine equation ax 3+by 3+cz 3=0. Acta Math. 85, 203–362 (1951)
Selmer, E.S.: Sufficient congruence conditions for the existence of rational points on certain cubic surfaces. Math. Scand. 1, 113–119 (1953)
Selmer, E.S.: The rational solutions of the Diophantine equation η 2=ξ 3−D for |D|≤100. Math. Scand. 4, 281–286 (1956)
Serre, J.-P.: Quelques applications du théoreme de densité de Chebotarev. Publ. Math. Inst. Hautes Études Sci. 54, 123–202 (1981) [[5661], vol. 3, pp. 563–641]
Serre, J.-P.: Lectures on the Mordell-Weil Theorem, 3rd ed. Vieweg, Wiesbaden (1989) 1997
Shelah, S.: Primitive recursive bounds for van der Waerden numbers. J. Am. Math. Soc. 1, 683–697 (1988)
Shepherd-Barron, N.I., Taylor, R.: mod 2 and mod 5 icosahedral representations. J. Am. Math. Soc. 10, 283–298 (1997)
Shimura, G.: Yutaka Taniyama and his time. Bull. Lond. Math. Soc. 21, 186–196 (1989)
Shimura, G., Taniyama, Y.: Complex Multiplication of Abelian Varieties and Its Applications to Number Theory. Math. Soc. Jpn., Tokyo (1961) [Expanded version: Shimura, G., Abelian Varieties with Complex Multiplication and Modular Functions, Princeton (1998)]
Shintani, T.: On certain ray class invariants of real quadratic fields. J. Math. Soc. Jpn. 30, 139–167 (1978)
Shorey, T.N., van der Poorten, A.J., Tijdeman, R., Schinzel, A.: Applications of the Gel’fond-Baker method to Diophantine equations. In: Transcendence Theory: Advances and Applications, pp. 59–77. Academic Press, San Diego (1977)
Šidlovskiĭ, A.B.: On a criterion for algebraic independence of values of a class of entire functions. Dokl. Akad. Nauk SSSR 100, 221–224 (1955) (in Russian)
Šidlovskiĭ, A.B.: On a criterion for algebraic independence of values of a class of entire functions. Izv. Akad. Nauk SSSR, Ser. Mat. 23, 35–66 (1959) (in Russian)
Šidlovskiĭ, A.B.: Transcendental Numbers. Nauka, Moscow (1987) (in Russian) [English translation: de Gruyter, 1989]
Siegel, C.L.: Approximation algebraischer Zahlen. Math. Z. 10, 173–213 (1921) [[5778], vol. 1, pp. 6–46]
Siegel, C.L.: Ueber den Thueschen Satz. Christiania Vid. Selsk, Skr., 1921, nr. 16 [[5778], vol. 1, pp. 103–112]
Siegel, C.L.: Darstellung total positiver Zahlen durch Quadrate. Math. Z. 11, 246–275 (1921) [[5778], vol. 1, pp. 47–76]
Siegel, C.L.: Additive Theorie der Zahlkörper, I. Math. Ann. 87, 1–35 (1922) [[5778], vol. 1, pp. 119–153]
Siegel, C.L.: Additive Theorie der Zahlkörper, II. Math. Ann. 88, 184–210 (1923) [[5778], vol. 1, pp. 180–206]
Siegel, C.L.: Über einige Anwendungen diophantischer Approximationen. Abh. Kgl. Preuß. Akad. Wiss. Berlin, 1929, 1–70 [[5778], vol. 1, pp. 209–266]
Siegel, C.L.: Über Riemanns Nachlaßzur analytischen Zahlentheorie. Quellen Stud. Gesch. Math., Astron. Phys. 2, 45–80 (1932) [[5778], vol. 1, pp. 275–310]
Siegel, C.L.: Die Gleichung ax n−by n=c. Math. Ann. 114, 57–68 (1937) [[5778], vol. 2, pp. 8–19]
Siegel, C.L.: Equivalence of quadratic forms. Am. J. Math. 63, 658–680 (1941) [[5778], vol. 2, pp. 217–239]
Siegel, C.L.: Transcendental Numbers. Ann. Math. Stud., vol. 16. Princeton University Press, Princeton (1949) [German translation: Transzendente Zahlen, Mannheim (1967)]
Siegel, C.L.: Indefinite quadratische Formen und Funktionentheorie. I. Math. Ann. 124, 17–54 (1951) [[5778], vol. 3, pp. 85–91]
Siegel, C.L.: Indefinite quadratische Formen und Funktionentheorie. II. Math. Ann. 124, 364–387 (1951) [[5778], vol. 3, pp. 85–91]
Siegel, C.L.: Normen algebraischer Zahlen. Nachr. Ges. Wiss. Göttingen, 1973, nr. 11, 197–215 [[5778], vol. 4, pp. 250–268]
Sikorav, J.-C.: Valeurs des formes quadratiques indéfinies irrationnelles (d’après G.A. Margulis). Prog. Math. 81, 307–315 (1990)
Silverman, J.H.: The Arithmetic of Elliptic Curves. Springer, Berlin (1986); 2nd ed. 2009 [Reprint: 1992]
Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Springer, Berlin (1994)
Skolem, Th.: Unlösbarkeit von Gleichungen, deren entsprechende Kongruenz für jeden Modul lösbar ist. Avh. Norske Vid. Akad. Oslo, 1942, nr. 4, 1–28
Skolem, Th.: Einige Bemerkungen über die Auffindung der rationalen Punkte auf gewissen algebraischen Gebilden. Math. Z. 63, 295–312 (1955)
Skorobogatov, A.N.: Beyond the Manin obstruction. Invent. Math. 135, 399–424 (1999)
Skorobogatov, A.[N.]: Torsors and Rational Points. Cambridge University Press, Cambridge (2001)
Skriganov, M.M.: On integer points in polygons. Ann. Inst. Fourier 43, 313–323 (1993)
Skriganov, M.M.: Ergodic theory on SL(n), Diophantine approximations and anomalies in the lattice point problem. Invent. Math. 132, 1–72 (1998)
Skriganov, M.M., Starkov, A.N.: On logarithmically small errors in the lattice point problem. Ergod. Theory Dyn. Syst. 20, 1469–1476 (2000)
Smith, H.J.S.: On the orders and genera of ternary quadratic forms. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 157, 255–298 (1867) [[5834], vol. 1, pp. 455–506]
Soifer, A.: Pierre Joseph Henry Baudet: Ramsey theory before Ramsey. Geombinatorics Q. 6, 60–70 (1996)
Soundararajan, K.: Mean-values of the Riemann zeta-function. Mathematika 42, 158–174 (1995)
Soundararajan, K.: Extreme values of zeta and L-functions. Math. Ann. 342, 467–486 (2008)
Soundararajan, K.: The distribution of smooth numbers in arithmetic progressions. In: Anatomy of Integers, pp. 115–128. Am. Math. Soc., Providence (2008)
Soundararajan, K.: Moments of the Riemann zeta-function. Ann. Math. 179, 981–993 (2009)
Speiser, A.: Die Theorie der Gruppen der endlichen Ordnung, 2nd ed. Springer, Berlin (1923) 1927, 3rd ed. 1937
Speiser, A.: Rudolf Fueter. Elem. Math. 5, 98–99 (1950)
Spencer, D.C.: On a Hardy-Littlewood problem of diophantine approximation. Proc. Camb. Philos. Soc. 35, 527–547 (1939)
Spencer, D.C.: The lattice points of tetrahedra. J. Math. Phys. MIT 21, 189–197 (1942)
Sprindžuk, V.G.: On the largest prime divisor of a binary form. Dokl. Akad. Nauk Belorus. 15, 389–391 (1971) (in Russian)
Sprindžuk, V.G.: Metric Theory of Diophantine Approximations. Nauka, Moscow (1977) (in Russian) [English translation: Wiley, 1979]
Stäckel, P.: Ueber Goldbachs empirisches Theorem: Jede grade Zahl kann als Summe von zwei Primzahlen dargestellt werden. Nachr. Ges. Wiss. Göttingen, 1896, 292–299
Stanley, G.K.: On the representation of a number as the sum of seven squares. J. Lond. Math. Soc. 2, 91–96 (1927)
Stanley, G.K.: On the representations of a number as the sum of squares and primes. J. Lond. Math. Soc. 3, 62–64 (1928)
Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980)
Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980)
Staś, W.: Über das Verhalten der Riemannschen ζ-Funktion und einiger verwandter Funktionen, in der Nähe der Geraden σ=1. Acta Arith. 7, 217–224 (1961/1962)
Staś, W.: On the order of Dedekind zeta-functions in the critical strip. Funct. Approx. Comment. Math. 4, 19–26 (1976)
von Sterneck, R.D.: On the distribution of quadratic residues and non-residues of a prime number. Mat. Sb. 20, 267–284 (1898) (in Russian)
Stevens, H.: On Jacobsthal’s g(n)-function. Math. Ann. 226, 95–97 (1977)
Stiemke, E.: Über unendliche algebraische Zahlkörper. Math. Z. 25, 9–39 (1926)
Straßmann, R.: Über den Wertevorrat von Potenzreihen im Gebiet der \(\mathfrak{p}\)-adischen Zahlen. J. Reine Angew. Math. 159, 13–28 (1928); add. 159, 65–66 (1928)
Strauch, O.: Some new criterions for sequences which satisfy Duffin–Schaeffer conjecture, I. Acta Math. Univ. Comen. 42/43, 87–95 (1983) (in Russian)
Strauch, O.: Some new criterions for sequences which satisfy Duffin–Schaeffer conjecture, II. Acta Math. Univ. Comen. 44/45, 55–65 (1984) (in Russian)
Strauch, O.: Some new criterions for sequences which satisfy Duffin–Schaeffer conjecture, III. Acta Math. Univ. Comen. 48/49, 37–50 (1986) (in Russian)
Stuhler, U.: Martin Kneser (21.1.1928–16.2.2004). Jahresber. Dtsch. Math.-Ver. 108, 45–61 (2006)
Suetuna, Z.: On the mean value of L-functions. Jpn. J. Math. 1, 69–82 (1924)
Suetuna, Z.: The zeros of L-functions on the critical line. Tohoku Math. J. 24, 313–331 (1925)
Suetuna, Z.: Über die approximative Funktionalgleichung für Dirichletsche L-Funktionen. Jpn. J. Math. 9, 111–116 (1932)
Swinnerton-Dyer, H.P.F.: Two special cubic surfaces. Mathematika 9, 54–56 (1962)
Swinnerton-Dyer, H.P.F.: The number of lattice points on a convex curve. J. Number Theory 6, 128–135 (1974)
Swinnerton-Dyer, H.P.F.: The solubility of diagonal cubic surfaces. Ann. Sci. Éc. Norm. Super. 34, 891–912 (2001)
Sylvester, J.J.: On the partition of an even number into two primes. Proc. Lond. Math. Soc. 4, 4–6 (1871) [[6014], vol. 2, pp. 709–711]
Szalay, L.: On the resolution of simultaneous Pell equations. Ann. Math. Inst. Inform. 34, 77–87 (2007)
Szekeres, G., Turán, P.: Über das zweite Hauptproblem der “Factorisatio Numerorum”. Acta Sci. Math. 6, 143–154 (1933) [[6227], vol. 1, pp. 1–12]
Szűsz, P.: Über einen Kusminschen Satz. Acta Math. Acad. Sci. Hung. 12, 447–453 (1961)
Takagi, T.: Über eine Theorie des relativ-Abelschen Zahlkörpers. J. Fac. Sci. Univ. Tokyo 41(9), 1–133 (1920)
Taketa, K.: Neuer Beweis eines Satzes von Herrn Furtwängler über die metabelschen Gruppen. Jpn. J. Math. 9, 199–218 (1932)
Tartakovskiĭ, V.A.: Auflösung der Gleichung x 4−ϱy 4=1. Bull. Acad. Sci. Leningrad 20, 301–324 (1926)
Tartakovskiĭ, V.A.: Die Gesamtheit der Zahlen, die durch eine positive quadratische Form F(x 1,x 2,…,x s ) (s≥4) darstellbar sind, I. Izv. Akad. Nauk SSSR, Ser. Mat. 2, 111–122 (1929)
Tartakovskiĭ, V.A.: Die Gesamtheit der Zahlen, die durch eine positive quadratische Form F(x 1,x 2,…,x s ) (s≥4) darstellbar sind, II. Izv. Akad. Nauk SSSR, Ser. Mat. 2, 165–196 (1929)
Tate, J.: Fourier analysis in number fields and Hecke’s zeta functions. Ph.D. thesis, Princeton Univ. (1950) [Reprint: [950], pp. 305–347]
Tate, J.: The arithmetic of elliptic curves. Invent. Math. 23, 179–206 (1974)
Tate, J.: Local constants. In: Algebraic Number Fields: L-functions and Galois Properties, pp. 89–131. Academic Press, San Diego (1977)
Tatuzawa, T.: The approximate functional equation for Dirichlet’s L-series. Jpn. J. Math. 22, 19–25 (1952)
Taussky-Todd, O.: Arnold Scholz zum Gedächtnis. Math. Nachr. 7, 379–386 (1952)
Taylor, A.D.: A note on van der Waerden’s theorem. J. Comb. Theory, Ser. A 33, 215–219 (1982)
Taylor, M.J.: Albrecht Fröhlich, 1916–2001. Bull. Lond. Math. Soc. 38, 329–350 (2006)
Taylor, R.: Icosahedral Galois representations. Pac. J. Math., 1997, Special Issue, 337–347
Taylor, R.: On icosahedral Artin representations, II. Am. J. Math. 125, 549–566 (2003)
Taylor, R.: Galois representations. Ann. Fac. Sci. Toulouse 13, 73–119 (2004)
Taylor, S.J.: Paul Lévy. Bull. Lond. Math. Soc. 7, 300–320 (1975)
Taylor, S.J.: Thomas Muirhead Flett. Bull. Lond. Math. Soc. 9, 330–339 (1977)
Tchudakoff, N.: On zeros of the function ζ(s). Dokl. Akad. Nauk SSSR, 1936, nr. 1, 201–204
Tchudakoff, N.: On the difference between two neighbouring prime numbers. Mat. Sb. 1, 799–814 (1936)
Tchudakoff, N.: On Goldbach-Vinogradov theorem. Ann. Math. 48, 515–545 (1947)
Tenenbaum, G.: Introduction à la théorie analytique et probabiliste des nombres. Institut Élie Cartan, Nancy (1990); 2nd ed., Paris, 1995 [English translation: Introduction to Analytic and Probabilistic Number Theory, Cambridge (1995)]
Tenenbaum, G.: Sur une question d’Erdős et Schinzel, II. Invent. Math. 99, 215–224 (1990)
Tenenbaum, G.: Cribler les entiers sans grand facteur premier. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 345, 377–384 (1993)
Thang, N.Q.: A note on the Hasse principle. Acta Arith. 54, 171–184 (1990)
Thue, A.: Computation of all solutions of certain equations of the form ax r−by r=f. Christiania Vid. Selsk, Skr., 1918, nr. 4, 1–9 (in Norwegian)
Thunder, J.L.: Siegel’s lemma for function fields. Mich. Math. J. 42, 147–162 (1995)
Titchmarsh, E.C.: The mean-value of the zeta-function on the critical line. Proc. Lond. Math. Soc. 27, 137–150 (1927)
Titchmarsh, E.C.: On an inequality satisfied by the zeta-function of Riemann. Proc. Lond. Math. Soc. 28, 70–80 (1928)
Titchmarsh, E.C.: On van der Corput’s method and the Zeta function of Riemann, II. Q. J. Math. 2, 313–320 (1931)
Titchmarsh, E.C.: On van der Corput’s method and the Zeta function of Riemann, III. Q. J. Math. 3, 133–141 (1932)
Titchmarsh, E.C.: On van der Corput’s method and the Zeta function of Riemann, V. Q. J. Math. 5, 195–210 (1934)
Titchmarsh, E.C.: The zeros of the Riemann zeta-function. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 151, 234–255 (1935)
Titchmarsh, E.C.: On the order of \(\zeta(\frac{1}{2}+it)\). Q. J. Math. 13, 11–17 (1942)
Tolev, D.I.: On the number of representations of an odd integer as a sum of three primes, one of which belongs to an arithmetic progression. Tr. Mat. Inst. Steklova 218, 415–432 (1997) (in Russian)
Trifonov, O.: On the squarefree problem, II. Math. Balk. 3, 284–295 (1989)
Tschebotareff, N.G.: Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebener Substitutionsklasse gehören. Math. Ann. 95, 191–228 (1926) (see also Čebotarev)
Tunnell, J.: Artin’s conjecture for representations of octahedral type. Bull. Am. Math. Soc. 5, 173–175 (1981)
Turing, A.M.: Some calculations of the Riemann zeta-function. Proc. Lond. Math. Soc. 3, 99–117 (1953)
Turjányi, S.: Eine Bemerkung zum Hilbert-Kamke-Problem. Publ. Math. (Debr.) 21, 89–93 (1974)
Turnbull, H.W.: Colin Maclaurin. Am. Math. Mon. 54, 318–322 (1947)
Uchida, K.: On Artin L-functions. Tohoku Math. J. 27, 75–81 (1975)
Valfiš, A.Z.: On the representation of integers by sums of squares. Asymptotical formulas. Usp. Mat. Nauk 7(6), 97–178 (1951) (in Russian)
Valson, C.A.: La vie et les travaux du baron Cauchy, vols. I–II, Gauthier-Villars, Paris (1868)
van de Lune, J., te Riele, H.J.J.: On the zeros of the Riemann zeta function in the critical strip, III. Math. Comput. 41, 759–767 (1983); corr. 46, 771 (1986)
van de Lune, J., te Riele, H.J.J.: Recent progress on the numerical verification of the Riemann hypothesis. CWI Newsletter, 1984, nr. 2, 35–37
van de Lune, J., te Riele, H.J.J., Winter, D.T.: On the zeros of the Riemann zeta function in the critical strip, IV. Math. Comput. 46, 667–681 (1986)
van den Dries, L.: Elimination theory for the ring of algebraic integers. J. Reine Angew. Math. 388, 189–205 (1988)
van den Dries, L., Macintyre, A.: The logic of Rumely’s local-global principle. J. Reine Angew. Math. 407, 33–56 (1990)
van der Corput, J.G.: Over roosterpunten in het platte vlak. (De beteekenis van de methoden van Voronoï en Pfeiffer.), pp. 1–128. Leiden, Groningen (1919)
van der Corput, J.G.: Über Gitterpunkte in der Ebene. Math. Ann. 81, 1–10 (1920)
van der Corput, J.G.: Zahlentheoretische Abschätzungen. Math. Ann. 84, 53–79 (1921)
van der Corput, J.G.: Verschärfung der Abschätzung beim Teilerproblem. Math. Ann. 87, 39–65 (1922); corr. 89, 160 (1923)
van der Corput, J.G.: Neue zahlentheoretische Abschätzungen, II. Math. Z. 29, 397–426 (1929)
van der Corput, J.G.: Zum Teilerproblem. Math. Ann. 98, 697–716 (1928)
van der Corput, J.G., Koksma, J.F.: Sur l’ordre de grandeur de la fonction ζ(s) de Riemann dans la bande critique. Ann. Fac. Sci. Toulouse 22, 1–39 (1930)
van der Waal, R.W.: On a conjecture of Dedekind on zeta-functions. Indag. Math. 37, 83–86 (1975)
van der Waerden, B.L.: Beweis einer Baudet’schen Vermutung. Nieuw Arch. Wiskd. 15, 212–216 (1927)
van der Waerden, B.L.: How the proof of Baudet’s conjecture was found. In: Studies in Pure Mathematics, pp. 251–260. Academic Press, San Diego (1971)
Vandiver, H.S.: On sets of three consecutive integers which are quadratic or cubic residues of primes. Bull. Am. Math. Soc. 31, 33–38 (1925)
van Lint, J.H., Richert, H.-E.: On primes in arithmetic progressions. Acta Arith. 11, 209–216 (1965)
Vaughan, R.C.: On the order of magnitude of Jacobsthal’s function. Proc. Edinb. Math. Soc. 20, 329–331 (1976/1977)
Vaughan, R.C.: On Waring’s problem for cubes. J. Reine Angew. Math. 365, 122–170 (1986)
Vaughan, R.C.: On Waring’s problem for cubes, II. J. Lond. Math. Soc. 39, 205–218 (1989)
Vaughan, R.C.: On Waring’s problem for smaller exponents. Proc. Lond. Math. Soc. 52, 445–463 (1986)
Vaughan, R.C.: On Waring’s problem for smaller exponents, II. Mathematika 33, 6–22 (1986)
Venkov, B.A., Natanson, I.P.: Obituary: Rodion Osievič Kuzmin (1891–1949). Usp. Mat. Nauk 4(4), 148–155 (1949) (in Russian)
Verebrusov, A.S.: On the number of solutions of cubic equations in two variables. Mat. Sb. 26, 115–129 (1907) (in Russian)
Vilčinskiĭ, V.T.: Rational approximations to almost all real numbers. Vesti AN BSSR, 1979, nr. 6, 20–24 (in Russian)
Vinogradov, I.M.: A new method to find asymptotical expression for arithmetical functions. Izv. Ross. Akad. Nauk, Ser. Mat. 11, 1347–1378 (1917) (in Russian)
Vinogradov, I.M.: On a general theorem of Waring. Mat. Sb. 31, 490–507 (1924) (in Russian)
Vinogradov, I.M.: Analytischer Beweis des Satzes über die Verteilung der Bruchteile eines Polynoms. Izv. Akad. Nauk SSSR, Ser. Mat. 21, 567–578 (1927)
Vinogradov, I.M.: On Waring’s theorem. Izv. Akad. Nauk SSSR, Ser. Mat. 1, 393–400 (1928) (in Russian)
Vinogradov, I.M.: On representations of a number by an integral polynomial in several variables. Izv. Akad. Nauk SSSR, Ser. Mat. 1, 401–414 (1928) (in Russian)
Vinogradov, I.M.: A new solution of Waring’s problem. Dokl. Akad. Nauk SSSR, 1934, nr. 2, 337–341 (in Russian)
Vinogradov, I.M.: On the upper bound for G(n) in Waring’s problem. Izv. Akad. Nauk SSSR, Ser. Mat. 7, 1455–1470 (1934) (in Russian)
Vinogradov, I.M.: On Waring’s problem. Ann. Math. 36, 395–405 (1935)
Vinogradov, I.M.: An asymptotic formula for the number of representations in Waring’s problem. Mat. Sb. 42, 531–534 (1935)
Vinogradov, I.M.: On asymptotic formula in Warings problem. Mat. Sb. 1, 169–174 (1936)
Vinogradov, I.M.: Sur quelques inégalités nouvelles de la théorie des nombres. C. R. Acad. Sci. Paris 202, 1361–1362 (1936)
Vinogradov, I.M.: A new method of estimation of trigonometric sums. Mat. Sb. 1, 175–188 (1936) (in Russian)
Vinogradov, I.M.: On the estimation of trigonometrical sums. Dokl. Akad. Nauk SSSR 34, 182–183 (1942) (in Russian)
Vinogradov, I.M.: A new estimate of the function ζ(1+it). Izv. Akad. Nauk SSSR, Ser. Mat. 22, 161–164 (1958) (in Russian)
Voskresenskiĭ, V.E., Malyšev, A.V., Perelmuter, G.I.: Nikolai Grigorevič Čudakov (on the occasion of his seventieth birthday). Usp. Mat. Nauk 30(3), 195–197 (1975) (in Russian)
Wakabayashi, I.: Number of solutions for cubic Thue equations with automorphisms. Ramanujan J. 14, 131–154 (2007)
Waldschmidt, M.: Les contributions de Serge Lang à la théorie des nombres transcendants. Math. Gaz. 108, 35–46 (2006)
Walfisz, A.: Zur Abschätzung von \(\zeta(\frac{1}{2}+it)\). Nachr. Ges. Wiss. Göttingen, 1924, 155–158
Walfisz, A.: Über die Wirksamkeit einiger Abschätzungen trigonometrischer Summen. Acta Arith. 4, 108–180 (1958)
Walsh, P.G.: Sharp bounds for the number of solutions to simultaneous Pell equations. Acta Arith. 126, 125–137 (2007)
Walter, W.: Das wissenschafliche Werk von Erich Kamke. Jahresber. Dtsch. Math.-Ver. 69, 193–205 (1968)
Waring, E.: Meditationes Algebraicae. Cambridge University Press, Cambridge (1770); 2nd ed. 1782 [English translation: Am. Math. Soc., 1991]
Waterhouse, W.C.: Pairs of quadratic forms. Invent. Math. 37, 157–164 (1976)
Waterhouse, W.C.: Pairs of symmetric bilinear forms in characteristic 2. Pac. J. Math. 69, 275–283 (1977)
Waterhouse, W.C.: A nonsymmetric Hasse-Minkowski theorem. Am. J. Math. 99, 755–759 (1977)
Watson, G.L.: Some problems in the theory of numbers. Ph.D. thesis, University College London (1953)
Watson, G.L.: On indefinite quadratic forms in three and four variables. J. Lond. Math. Soc. 28, 239–242 (1953)
Watson, G.L.: Representation of integers by indefinite quadratic forms. Mathematika 2, 32–38 (1955)
Watson, G.L.: Distinct small values of quadratic forms. Mathematika 7, 36–40 (1960)
Watson, G.L.: Indefinite quadratic polynomials. Mathematika 7, 141–144 (1960)
Watson, G.L.: Indefinite quadratic Diophantine equations. Mathematika 8, 32–38 (1961)
Watson, G.L.: Quadratic Diophantine equations. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 253, 227–254 (1960/1961)
Watt, N.: Exponential sums and the Riemann zeta function, II. J. Lond. Math. Soc. 39, 385–404 (1989)
Watt, N.: Short intervals almost all containing primes. Acta Arith. 72, 131–167 (1995)
Weber, H.: Lehrbuch der Algebra. vols. I–III. Vieweg, Braunschweig (1894–1908)
Weber, H.: Über Zahlengruppen in algebraischen Körpern. Math. Ann. 48, 433–437 (1897)
Weber, H.: Über Zahlengruppen in algebraischen Körpern, II. Math. Ann. 49, 83–100 (1897)
Weber, H.: Über Zahlengruppen in algebraischen Körpern, III. Math. Ann. 50, 1–26 (1898)
Wedeniwski, S.: Results connected with the first 100 billion zeros of the Riemann zeta function. http://www.zetagrid.net/zeta/math/zeta.result.100billion.zeros.html
Weierstrass, C.: Zu Lindemann’s Abhandlung “Über die Ludolphsche Zahl”. SBer. Kgl. Preuß. Akad. Wiss. Berlin, 1885, 1067–1085. [Mathematische Werke, vol. 2, pp. 341–362, Berlin (1895)]
Weil, A.: L’arithmétique sur les courbes algébriques. Acta Math. 52, 281–315 (1928) [[6631], vol. 1, pp. 11–35]
Weil, A.: Sur un théorème de Mordell. Bull. Sci. Math. 54, 182–191 (1929) [[6631], vol. 1, pp. 47–56]
Weil, A.: Remarques sur des résultats récents de C. Chevalley. C. R. Acad. Sci. Paris 203, 1208–1210 (1936) [[6631], vol. 1, pp. 145–146]
Weil, A.: Basic Number Theory. Springer, Berlin (1967); 2nd ed. 1973 [Reprint: 1975], 3rd ed. 1974
Western, A.E.: Note on the magnitude of the difference between successive primes. J. Lond. Math. Soc. 9, 276–278 (1934)
Weston, T.: Kummer theory of abelian varieties and reductions of Mordell-Weil groups. Acta Arith. 110, 77–88 (2003)
Westzynthius, E.: Sur la distribution des entiers qui ne sont divisibles par aucun parmi les n plus petits nombres premiers. C. R. Acad. Sci. Paris 193, 805–807 (1931)
Westzynthius, E.: Über die Verteilung der Zahlen, die zu der n ersten Primzahlen teilerfremd sind. Commun. Phys. Math. Helsingfors, (5) 25, 1–37 (1931)
Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352 (1916)
Weyl, H.: Zur Abschätzung von ζ(1+it). Math. Z. 10, 88–101 (1921)
Weyl, H.: Bemerkung zur Hardy-Littlewoodschen Lösung des Waringschen Problems. Nachr. Ges. Wiss. Göttingen, 1922, 189–192
Wieferich, A.: Über die Darstellung der Zahlen als Summen von Biquadraten. Math. Ann. 66, 106–108 (1909)
Willerding, M.F.: Determination of all classes of positive quaternary quadratic forms which represent all (positive) integers. Bull. Am. Math. Soc. 54, 334–337 (1948)
Wilson, B.M.: An application of Pfeiffer’s method to a problem of Hardy and Littlewood. Proc. Lond. Math. Soc. 22, 248–253 (1923)
Wirsing, E.: On the theorem of Gauss-Kusmin-Levy and a Frobenius-type theorem for function spaces. Acta Arith. 24, 507–528 (1974)
Witt, E.: Theorie der quadratischen Formen in beliebigen Körpern. J. Reine Angew. Math. 176, 31–44 (1937)
Wong, S.: Power residues on abelian varieties. Manuscr. Math. 102, 129–138 (2000)
Wooley, T.D.: Large improvements in Waring’s problem. Ann. Math. 135, 131–164 (1992)
Wooley, T.D.: On Vinogradov’s mean value theorem. Mathematika 39, 379–399 (1992)
Wooley, T.D.: On Vinogradov’s mean value theorem, II. Mich. Math. J. 40, 175–180 (1993)
Wooley, T.D.: The application of a new mean value theorem to the fractional parts of polynomials. Acta Arith. 65, 163–179 (1993)
Wooley, T.D.: On exponential sums over smooth numbers. J. Reine Angew. Math. 488, 79–140 (1997)
Wooley, T.D.: Vinogradov’s mean value theorem via efficient congruencing, to appear. arXiv:1101.0574
Wyman, M.: Leo Moser (1921–1970). Can. Math. Bull. 15, 1–4 (1972)
Xu, Y., Yau, S.S.-T.: A sharp estimate of the number of integral points in a tetrahedron. J. Reine Angew. Math. 423, 199–219 (1992)
Xu, Y., Yau, S.S.-T.: A sharp estimate of the number of integral points in a 4-dimensional tetrahedra. J. Reine Angew. Math. 473, 1–23 (1996)
Xuan, T.Z.: Integers with no large prime factors. Acta Arith. 69, 303–327 (1995)
Young, J., Potler, A.: First occurrence prime gaps. Math. Comput. 52, 221–224 (1989)
Yu, H.B.: On Waring’s problem with polynomial summands. Acta Arith. 76, 131–144 (1996)
Yu, H.B.: On Waring’s problem with polynomial summands, II. Acta Arith. 86, 245–254 (1998)
Yu, H.B.: On Waring’s problem with quartic polynomial summands. Acta Arith. 80, 77–82 (1997)
Yu, H.B.: On Waring’s problem with quartic polynomial summands, II. J. China Univ. Sci. Tech. 28, 629–635 (1998)
Yuan, P.Z.: On the number of solutions of simultaneous Pell equations. Acta Arith. 101, 215–221 (2002)
Yuan, P.Z.: On the number of solutions of x 2−4m(m+1)y 2=y 2−bz 2−1. Proc. Am. Math. Soc. 132, 1561–1566 (2004)
Zaharescu, A.: Small values of n 2 α (mod 1). Invent. Math. 121, 379–388 (1995)
Zavorotnyĭ, N.I.: On the fourth moment of the Riemann zeta function. In: Automorphic Functions and Number Theory, Vladivostok, pp. 69–124 (1989) (in Russian)
Zulauf, A.: Beweis einer Erweiterung des Satzes von Goldbach-Vinogradov. J. Reine Angew. Math. 190, 169–198 (1952)
Zulauf, A.: Über die Darstellung natürlichen Zahlen als Summen von Primzahlen aus gegebenen Restklassen und Quadraten mit gegebenen Koeffizienten, I. J. Reine Angew. Math. 192, 210–229 (1953)
Zulauf, A.: Über die Darstellung natürlichen Zahlen als Summen von Primzahlen aus gegebenen Restklassen und Quadraten mit gegebenen Koeffizienten, II. J. Reine Angew. Math. 193, 39–53 (1954)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag London Limited
About this chapter
Cite this chapter
Narkiewicz, W. (2012). The Twenties. In: Rational Number Theory in the 20th Century. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-0-85729-532-3_3
Download citation
DOI: https://doi.org/10.1007/978-0-85729-532-3_3
Publisher Name: Springer, London
Print ISBN: 978-0-85729-531-6
Online ISBN: 978-0-85729-532-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)