The Zeta Function of an Algebraic Number Field and Some Applications

  • Steve Wright
Part of the Lecture Notes in Mathematics book series (LNM, volume 2171)


At the end of Sect.  4.6 of Chap.  4, we left ourselves with the problem of determining the finite nonempty subsets S of the positive integers such that for infinitely many primes p, S is a set of non-residues of p. We observed there that if S has this property then the product of all the elements in every subset of S of odd cardinality is never a square. The object of this chapter is to prove the converse of this statement, i.e., we wish to prove Theorem  4.12. The proof of Theorem  4.12 that we present uses ideas that are closely related to the ones that Dirichlet used in his proof of Theorem 4.5, together with some technical improvements due to Hilbert. The key tool that we need is an analytic function attached to algebraic number fields, called the zeta function of the field. The definition of this function requires a significant amount of mathematical technology from the theory of algebraic numbers, and so in Sect. 5.1 we begin with a discussion of the results from algebraic number theory that will be required, with Dedekind’s Ideal Distribution Theorem as the final goal of this section. The zeta function of an algebraic number field is defined and studied in Sect. 5.2; in particular, the Euler-Dedekind product formula for the zeta function is derived here. In Sect. 5.3 a product formula for the zeta function of a quadratic number field that will be required in the proof of Theorem  4.12 is derived from the Euler-Dedekind product formula. The proof of Theorem  4.12, the principal object of this chapter, is carried out in Sect. 5.4 and some results which are closely related to that theorem are also established there. In the interest of completeness, we prove in Sect. 5.5 the Fundamental Theorem of Ideal Theory, Theorem  3.16 of Chap.  3, since it is used in an essential way in the derivation of the Euler-Dedekind product formula.


  1. 1.
    B. Berndt, Classical theorems on quadratic residues. Enseignement Math. 22, 261–304 (1976)MathSciNetzbMATHGoogle Scholar
  2. 2.
    H. Cohen, Number Theory, vol. I (Springer, New York, 2000)zbMATHGoogle Scholar
  3. 3.
    J.B. Conway, Functions of One Complex Variable, vol. 1 (Springer, New York, 1978)CrossRefGoogle Scholar
  4. 4.
    K.L. Chung, A Course in Probability Theory (Academic Press, New York, 1974)zbMATHGoogle Scholar
  5. 5.
    H. Davenport, On character sums in finite fields. Acta Math. 71, 99–121 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    H. Davenport, Multiplicative Number Theory (Springer, New York, 2000)zbMATHGoogle Scholar
  7. 7.
    H. Davenport, P. Erdös, The distribution of quadratic and higher residues. Publ. Math. Debrecen 2, 252–265 (1952)MathSciNetzbMATHGoogle Scholar
  8. 8.
    R. Dedekind, Sur la Th \(\acute{\text{e}}\) orie des Nombres Entiers Alg \(\acute{\text{e}}\) briques (1877); English translation by J. Stillwell (Cambridge University Press, Cambridge, 1996)Google Scholar
  9. 9.
    P.G.L. Dirichlet, Sur la convergence des series trigonom\(\acute{\text{e}}\) trique qui servent \(\grave{\text{a}}\) repr\(\acute{\text{e}}\) senter une fonction arbitraire entre des limites donn\(\acute{\text{e}}\) e. J. Reine Angew. Math. 4, 157–169 (1829)Google Scholar
  10. 10.
    P.G.L. Dirichlet, Beweis eines Satzes da\(\ss \) jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind, unendlich viele Primzahlen enh\(\ddot{\text{a}}\) lt. Abh. K. Preuss. Akad. Wiss. 45–81 (1837)Google Scholar
  11. 11.
    P.G.L. Dirichlet, Recherches sur diverses applications de l’analyse infinit\(\acute{\text{e}}\) simal \(\grave{\text{a}}\) la th\(\acute{\text{e}}\) orie des nombres. J. Reine Angew. Math. 19, 324–369 (1839); 21 (1–12), 134–155 (1840)Google Scholar
  12. 12.
    P.G.L. Dirichlet, Vorlesungen über Zahlentheorie (1863); English translation by J. Stillwell (American Mathematical Society, Providence, 1991)Google Scholar
  13. 13.
    J. Dugundji, Topology (Allyn and Bacon, Boston, 1966)zbMATHGoogle Scholar
  14. 14.
    P. Erdös, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U.S.A. 35, 374–384 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    L. Euler, Theoremata circa divisores numerorum in hac forma pa 2 ± qb 2 contentorum. Comm. Acad. Sci. Petersburg 14, 151–181 (1744/1746)Google Scholar
  16. 16.
    L. Euler, Theoremata circa residua ex divisione postestatum relicta. Novi Commet. Acad. Sci. Petropolitanea 7, 49–82 (1761)Google Scholar
  17. 17.
    L. Euler, Observationes circa divisionem quadratorum per numeros primes. Opera Omnia I-3, 477–512 (1783)Google Scholar
  18. 18.
    M. Filaseta, D. Richman, Sets which contain a quadratic residue modulo p for almost all p. Math. J. Okayama Univ. 39, 1–8 (1989)Google Scholar
  19. 19.
    C.F. Gauss, Disquisitiones Arithmeticae (1801); English translation by A. A. Clarke (Springer, New York, 1986)Google Scholar
  20. 20.
    C.F. Gauss, Theorematis arithmetici demonstratio nova. Göttingen Comment. Soc. Regiae Sci. XVI, 8 pp. (1808)Google Scholar
  21. 21.
    C.F. Gauss, Summatio serierum quarundam singularium. Göttingen Comment. Soc. Regiae Sci. 36 pp. (1811)Google Scholar
  22. 22.
    C.F. Gauss, Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et amplicationes novae, 1818, Werke, vol. II (Georg Olms Verlag, Hildescheim, 1973), pp. 47–64Google Scholar
  23. 23.
    C.F. Gauss, Theorematis fundamentallis in doctrina residuis demonstrationes et amplicationes novae. Göttingen Comment. Soc. Regiae Sci. 4, 17 pp. (1818)Google Scholar
  24. 24.
    C.F. Gauss, Theoria residuorum biquadraticorum: comentatio prima. Göttingen Comment. Soc. Regiae Sci. 6, 28 pp. (1828)Google Scholar
  25. 25.
    C.F. Gauss, Theoria residuorum biquadraticorum: comentatio secunda. Göttingen Comment. Soc. Regiae Sci. 7, 56 pp. (1832)Google Scholar
  26. 26.
    D. Gröger, Gauß’ Reziprozitätgesetze der Zahlentheorie: Eine Gesamtdarstellung der Hinterlassenschaft in Zeitgemäßer Form (Erwin-Rauner Verlag, Augsburg, 2013)Google Scholar
  27. 27.
    E. Hecke, Vorlesungen über die Theorie der Algebraischen Zahlen (1923); English translation by G. Brauer and J. Goldman (Springer, New York, 1981)zbMATHGoogle Scholar
  28. 28.
    D. Hilbert, Die Theorie der Algebraischen Zahlkörper (1897); English translation by I. Adamson (Springer, Berlin, 1998)Google Scholar
  29. 29.
    T. Hungerford, Algebra (Springer, New York, 1974)zbMATHGoogle Scholar
  30. 30.
    K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory (Springer, New York, 1990)CrossRefzbMATHGoogle Scholar
  31. 31.
    P. Kurlberg, The distribution of spacings between quadratic residues II. Isr. J. Math. 120, 205–224 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    P. Kurlberg, Z. Rudnick, The distribution of spacings between quadratic residues. Duke Math. J. 100, 211–242 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    J.L. Lagrange, Probl\(\grave{\text{e}}\) mes ind\(\acute{\text{e}}\) termin\(\acute{\text{e}}\) s du second degr\(\acute{\text{e}}\). Mém. Acad. R. Berlin 23, 377–535 (1769)Google Scholar
  34. 34.
    J.L. Lagrange, Reserches d’Arithm\(\acute{\text{e}}\) tique, 2nde partie. Nouv. Mém. Acad. Berlin 349–352 (1775)Google Scholar
  35. 35.
    E. Landau, Elementary Number Theory. English translation by J. Goodman (Chelsea, New York, 1958)zbMATHGoogle Scholar
  36. 36.
    A. Legendre, Reserches d’analyse indéterminée. Histoiré de l’Académie Royale des Sciences de Paris (1785), pp. 465–559, Paris 1788Google Scholar
  37. 37.
    A. Legendre, Essai sur la Th \(\acute{\text{e}}\) orie des Nombres (Paris, 1798)Google Scholar
  38. 38.
    F. Lemmermeyer, Reciprocity Laws (Springer, New York/Berlin/Heidelberg, 2000)CrossRefzbMATHGoogle Scholar
  39. 39.
    W.J. LeVeque, Topics in Number Theory, vol. II (Addison-Wesley, Reading, 1956)zbMATHGoogle Scholar
  40. 40.
    D. Marcus, Number Fields (Springer, New York, 1977)CrossRefzbMATHGoogle Scholar
  41. 41.
    H. Montgomery, R. Vaughan, Multiplicative Number Theory I: Classical Theory (Cambridge University Press, Cambridge, 2007)zbMATHGoogle Scholar
  42. 42.
    R. Nevenlinna, V. Paatero, Introduction to Complex Analysis (Addison-Wesley, Reading, 1969)Google Scholar
  43. 43.
    O. Ore, Les Corps Alg \(\acute{\text{e}}\) briques et la Th \(\acute{\text{e}}\) orie des Id \(\acute{\text{e}}\) aux (Gauthier-Villars, Paris, 1934)Google Scholar
  44. 44.
    G. Perel’muter, On certain character sums. Usp. Mat. Nauk. 18, 145–149 (1963)MathSciNetzbMATHGoogle Scholar
  45. 45.
    C. de la Vall\(\acute{\text{e}}\) e Poussin, Recherches analytiques sur la th\(\acute{\text{e}}\) orie des nombres premiers. Ann. Soc. Sci. Bruxelles 20, 281–362 (1896)Google Scholar
  46. 46.
    H. Rademacher, Lectures on Elementary Number Theory (Krieger, New York, 1977)zbMATHGoogle Scholar
  47. 47.
    G.F.B. Riemann, \(\ddot{\text{U}}\) ber die Anzahl der Primzahlen unter einer gegebenen Gr\(\ddot{\text{o}}\ss \) e. Monatsberischte der Berlin Akademie (1859), pp. 671–680Google Scholar
  48. 48.
    K. Rosen, Elementary Number Theory and Its Applications (Pearson, Boston, 2005)Google Scholar
  49. 49.
    J. Rosenberg, Algebraic K-Theory and Its Application (Springer, New York, 1996)Google Scholar
  50. 50.
    W. Schmidt, Equations over Finite Fields: an Elementary Approach (Springer, Berlin, 1976)CrossRefzbMATHGoogle Scholar
  51. 51.
    A. Selberg, An elementary proof of Dirichlet’s theorem on primes in arithmetic progressions, Ann. Math. 50, 297–304 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    A. Selberg, An elementary proof of the prime number theorem. Ann. Math. 50, 305–313 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    A. Shamir, Identity-based cryptosystems and signature schemes, in Advances in Cryptology, ed. by G.R. Blakely, D. Chaum (Springer, Berlin, 1985), pp. 47–53CrossRefGoogle Scholar
  54. 54.
    J. Shohat, J.D. Tamarkin, The Problem of Moments (American Mathematical Society, New York, 1943)CrossRefzbMATHGoogle Scholar
  55. 55.
    R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141, 553–572 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    J. Urbanowicz, K.S. Williams, Congruences for L-Functions (Kluwer, Dordrecht, 2000)CrossRefzbMATHGoogle Scholar
  57. 57.
    A. Weil, Sur les Courbes Algébriques et les Variétes qui s’en Déduisent (Hermann et Cie, Paris, 1948)zbMATHGoogle Scholar
  58. 58.
    A. Weil, Basic Number Theory (Springer, New York, 1973)CrossRefzbMATHGoogle Scholar
  59. 59.
    L. Weisner, Introduction to the Theory of Equations (MacMillan, New York, 1938)zbMATHGoogle Scholar
  60. 60.
    A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. Math. 141, 443–551 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    S. Wright, Quadratic non-residues and the combinatorics of sign multiplication. Ars Combin. 112, 257–278 (2013)MathSciNetzbMATHGoogle Scholar
  62. 62.
    S. Wright, Quadratic residues and non-residues in arithmetic progression. J. Number Theory 133, 2398–2430 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    S. Wright, On the density of primes with a set of quadratic residues or non-residues in given arithmetic progression, J. Combin. Number Theory 6, 85–111 (2015)MathSciNetzbMATHGoogle Scholar
  64. 64.
    B.F. Wyman, What is a reciprocity law? Am. Math. Monthly 79, 571–586 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    A. Zygmund, Trigonometric Series (Cambridge University Press, Cambridge, 1968)zbMATHGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Steve Wright
    • 1
  1. 1.Department of Mathematics and StatisticsOakland UniversityRochesterUSA

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