Abstract
A wide variety of questions of Harmonic Analysis arise naturally in various contexts of Analytic Number Theory; in what follows we consider a number of examples of this type.
The author is grateful to Dr. Ulrike Vorhauer for advice and assistance at all stages of preparation of this paper.
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References
G. F. Bachelis, On the upper and lower majorant properties in L p(G), Quart. J. Math. (Oxford) (2) 24 (1973), 119–128.
J. T. Barton, H. L. Montgomery and J. D. Vaaler, Note on a Diophantine inequality in several variables, to appear.
J. Beck, The modulus of polynomials with zeros on the unit circle: a problem of Erdös, Ann. of Math. (2) 134 (1991), 609–651.
J. Beck and W. W. L. Chen, Irregularities of distribution, Cambridge Tract 89, Cambridge University Press, Cambridge, 1987.
A. Beurling, Sur les intégrales de Fourier absolument convergentes et leur application á une transformation fonctionelle, Neuviéme congrés des mathématiciens Scandinaves, Helsingfors, 1938
E. Bombieri and H. Iwaniec, On the order of £(1/2 + it), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), 449–472.
J. Bourgain, On the distribution ofDirichlet sums, J. Anal. Math. 60 (1993), 21–32.
W. W. L. Chen, On irregularities of distribution, I, Mathematika 27 (1980), 153–170.
P. Erdös and P. Turân, On a problem in the theory of uniform distribution I, Nederl. Akad. Wetensch. Proc. 51 (1948), 1146–1154; II, 1262-1269, (= Indag. Math. 10, 370-378; 406-413).
P. X. Gallagher, The large sieve, Mathematika 14 (1967), 14–20.
-, On the distribution of primes in short intervals, Mathematika 23 (1976), 4–9.
D. A. Goldston and H. L. Montgomery, On pair correlations of zeros and primes in short intervals, An-alytic number theory and Diophantine problems (Stillwater, 1984), Birkhäuser Verlag, Boston-Basel-Berlin, 1987, 183–203.
G. Halâsz, On Roth’s method in the theory of irregularities of point distributions, Recent Progress in Analytic Number Theory (Durham, 1979), Vol. 2, Academic Press, London, 1981, 79–84.
J. H. Halton, On the efficiency of certain quasirandom sequences of points in evaluating multidimensional integrals, Num. Math. 2 (1960), 84–90.
C. S. Herz, Fourier transforms related to convex sets, Ann. of Math. (2) 75 (1961), 81–92.
C. S. Herz, On the number of lattice points in a convex set, Amer. J. Math. 84 (1962), 126–133.
M. N. Huxley, Area, Lattice Points and Exponential Sums, Clarendon Press, Oxford, 1996.
Ju. V. Linnik, The large sieve, Dokl. Akad. Nauk SSSR 30 (1941), 292–294.
H. L. Montgomery, The analytic principle of the large sieve, Bull. Amer. Math. Soc. 84 (1978), 547–567.
H. L. Montgomery, Irregularities of distribution, Congress of Number Theory (Zarautz, 1984), Universi-dad del Pais Vasco, Bilbao, 1989, 11–27.
H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS No. 84, Amer. Math. Soc, Providence, 1994.
H. L. Montgomery and J. D. Vaaler, A further generalization of Hubert’s inequality, Mathematika 45 (1999), 35–39.
H. L. Montgomery and R. C. Vaughan, Hubert’s inequality, J. London Math. Soc. (2) 8 (1974), 73–82.
H. L. Montgomery and R. C. Vaughan —, On the distribution of reduced residues, Annals of Math. 123 (1986), 311–333.
H. L. Montgomery and K. Soundararajan, Beyond pair correlation, to appear.
H. L. Montgomery and K. Soundararajan, Primes in short intervals, to appear.
A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen I, Abh. Math. Sem. Ham-burg 1 (1922), 77–98; II, 1 (1922), 250-251; III, 41 (1926), 224.
A.Rényi, Un nouveau théorème concernant les fonctions indépendantes et ses applications à la théorie des nombres, J. Math. Pures Appl. 28 (1949), 137–149.
K. F. Roth, On irregularities of distribution, Mathematika 1 (1954), 73–79.
K. F. Roth, On the large sieves of Linnik and Rényi, Mathematika 12 (1965), 1–9.
W. M. Schmidt, Irregularities of distribution, VII, Acta Arith. 21 (1972), 49–50.
A. Selberg, Collected Papers, Volume II, Springer-Verlag, Berlin, 1991.
P. Turán, On a new method of analysis and its applications, Wiley-Interscience, New York, 1984.
J. D. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. 12 (1985), 183–216.
G. S. Wagner, On a problem of Erdös in Diophantine approximation, Bull. London Math. Soc. 12 (1980), 81–88.
H. Weyl, Über ein Problem aus dem Gebiete der diophantischen Approximationen, Nachr. Ges. Wiss. Göttingen, Math.-phys. Kl. (1914), 234–244; Gesammelte Abhandlungen, Band I, Springer-Verlag, Berlin-Heidelberg-New York, 1968,487-497.
Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352; Gesammelte Abhandlungen, Band I, Springer-Verlag, Berlin-Heidelberg-New York, 1968, 563-599; Selecta Hermann Weyl, Birkhäuser Verlag, Basel-Stuttgart, 1956,111-147.
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Montgomery, H.L. (2001). Harmonic Analysis as found in Analytic Number Theory. In: Byrnes, J.S. (eds) Twentieth Century Harmonic Analysis — A Celebration. NATO Science Series, vol 33. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0662-0_13
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