Skip to main content
SpringerLink
Log in

The Classical Theory of Zeta and L-Functions

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

This expository paper is devoted to basic facts about Zeta and L-functions, following to some extent the historical development of the subject.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. N.H. Abel, Œuvres complètes de Niels Henrik Abel. Tome I. Contenant les mémoirs publiés par Abel. Edited and with a preface by L. Sylow and S. Lie. Imprimerie de Grøndahl & Son, Christiania; distributed by the Norwegian Mathematical Society, Oslo, 1981. viii+621 pp.

  2. Apéry R.: Irrationalité de ζ2 et ζ3. Astérisque 61, 11–13 (1979)

    MATH  Google Scholar 

  3. Ball K., Rivoal T.: Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs. Invent. Math. 146, 193–207 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Bernoulli, Ars conjectandi. Basel 1713.

  5. Bohr H., Jessen B.: Über die Werteverteilung der Riemannschen Zetafunktion. Acta Math. 58, 1–55 (1932)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bohr H., Landau E.: Sur les zéros de la fonction ζ(s) de Riemann. Comptes Rendus Acad. Sci. Paris 158, 106–110 (1914)

    MATH  Google Scholar 

  7. E. Bombieri, Remarks on Weil’s quadratic functional in the theory of prime numbers, I, Mem. Accad. Naz. Lincei, (2001), 48 pp.

  8. Bombieri E., Hejal D.A.: On the distribution of zeros of linear combination of Euler products. Duke Math. Journal 80, 851–862 (1995)

    Article  Google Scholar 

  9. E. Bombieri, J.C. Lagarias, Complements to Li’s criterion for the Riemann hypothesis, J. Number Theory, 77 (1999), 274–265.

    Google Scholar 

  10. Bombieri E., Mueller J.: On the zeros of certain Epstein zeta functions. Forum Math. 20, 359–385 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Borchsenius V., Jessen B.: Mean motions and values of the Riemann zeta function. Acta Math. 80, 97–166 (1948)

    Article  MathSciNet  Google Scholar 

  12. J. Bourgain, On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function, Int. Math. Res. Notices, (2000), 133–146.

  13. Burnol J.-F.: Sur les formules explicites. I. Analyse invariante. C. R. Acad. Sci. Paris Sér. I Math. 331, 423–428 (2000)

    MATH  MathSciNet  Google Scholar 

  14. F. Carlson, Über die Nullstellen der Dirichletschen Reihen und der Riemannschen ζ-funktion, Arkiv för Mat. Astr. och Fysik, 15 (1920), No. 20.

  15. Cassels J.W.S.: Footnote to a note of Davenport and Heilbronn. J. London Math. Soc. 36, 177–184 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  16. G. Castelnuovo, Memorie Scelte. Zanichelli Ed., Bologna 1937–XV.

  17. A. Cauchy, Sur une loi de reciprocité qui existe entre certaines fonctions, Bull. de la Societé Philomatique, (1817), 121–124. Also Œuvres. Sér. II, T. II, Gauthiers-Villars, Paris 1958.

  18. Conrey J.B.: More than two fifths of the zeros of the Riemann zeta function are on the critical line. J. reine Angew. Math. 399, 1–26 (1989)

    MATH  MathSciNet  Google Scholar 

  19. J.B. Conrey, A. Ghosh, A conjecture for the sixth power moment of the Riemann zetafunction, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), 35–59, Univ. Salerno, Salerno, 1992.

  20. Conrey J.B., Gonek S.M.: High moments of the Riemann zeta-function. Duke Math. J. 107, 577–604 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Davenport H., Heilbronn H.: On the zeros of certain Dirichlet series. J. London Math. Soc. 11, 181–185 (1936)

    Article  MATH  Google Scholar 

  22. Davenport H., Heilbronn H.: On the zeros of certain Dirichlet series (second paper). J. London Math. Soc. 11, 307–312 (1936)

    Article  MATH  Google Scholar 

  23. Deligne P., Goncharov A.B.: Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. (4) 38, 1–56 (2005)

    MATH  MathSciNet  Google Scholar 

  24. G. Lejeune Dirichlet, Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abh. der König. Akad. der Wissen. zu Berlin (1837), 45–71. Also Werke, Bd. 1 (1889), 313-342.

  25. Dueñez E., Miller S.J.: The effect of convolving families of L-functions on the underlying group of symmetries. Proc. London Math. Soc., (3) 99, 787–820 (2009)

    Article  MATH  Google Scholar 

  26. L. Euler, Institutiones Calculi Differentialis. Petropoli 1755. Also Opera Omnia, Ser. I, T. X, Teubner, Leipzig & Berlin 1913.

  27. L. Euler, Opuscula Analytica. 1785. Also Opera Omnia, Ser. I, T. III, Teubner, Geneva 1941.

  28. L. Euler, Introductio in Analysin Infinitorum. Bd. 1, Bousquet, Lausanne 1748. Also Opera Omnia, Ser. I, T. VIII, Teubner, Leipzig & Berlin 1922.

  29. L. Euler, Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques (lu en 1749). Histoire de l’Académie Royale des Sciences et Belles-Lettres, 17 (1761), 83–106; Berlin 1768. Also Opera Omnia, Ser. I, T. XV, Teubner, Leipzig & Berlin 1927.

  30. L. Euler, Variae observationes circa series infinitas, Novi commentarii academiae scientiarum Petropolitanae, 8 (1737) 1740, 160-188. Also Opera Omnia, Series I, T. XIV, Commentationes Analyticae, T. I, Lipsia et Berolini 1917, pp. 243-244.

  31. L. Euler, De numeris primis valde magnis, Novi commentarii academiae scientiarum Petropolitanae 9 (1762/3) 1764, 99-153. Also Opera Omnia, Series I,Vol. III, Commentationes Arithmeticae, Teubner, Lipsia et Berolini 1917, pp. 3-4.

  32. J.H. Fuss, Correspondance mathématique et physique de quelques célèbres géomètres du XVIII e siècle. St. Petersbourg 1843. Also Johnson Reprint Co. New York & London 1968.

  33. Grothendieck A.: Sur une note de Mattuck–Tate. J. reine Angew. Math. 208, 208–215 (1958)

    MathSciNet  Google Scholar 

  34. Guinand A.P.: Summation formulae and self-reciprocal functions, III. Quart. J. Math. Oxford 13, 30–39 (1942)

    Article  MathSciNet  Google Scholar 

  35. C. Hermite, Note au sujet de la communication de M. Stieltjes sur une fonction uniforme, Comptes Rendus Acad. Sci. Paris, 101 (1885), 112–115. Also Œuvres, T. IV, 184–187. Gauthier-Villars, Paris 1917.

  36. Hurwitz A.: Einige Eigenschaften der Dirichlet’schen Functionen \({F(s) = \sum\left(\frac{D}{n}\right)\frac{1}{n^s}}\) die bei der Bestimmung der Classenanzahlen binärer quadratischer Formen auftreten. Zeitschrift für Math. und Physik XXVII, 86–101 (1882)

    Google Scholar 

  37. N.M. Katz, P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Providence, RI, 1999. xii+419 pp.

  38. Keating J.P., Snaith N.C.: Random matrix theory and ζ(1/2 + it). Comm. Math. Phys. 214, 57–89 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  39. Yoonbok Lee, On the zeros of Epstein zeta functions, preprint 2009. leeyb@yonsei.ac.kr.

  40. A.M. Legendre, Exercices de calcul intégral. Paris 1814.

  41. Levinson N.: More than one third of zeros of Riemann’s zeta-function are on σ = 1/2. Advances in Math. 13, 383–436 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  42. Malmstén C.J.: De integralibus quibusdam definitis, seriebusque infinitis. J. reine angew. Math. 38, 1–39 (1849)

    MATH  Google Scholar 

  43. W. Narkiewicz, The Development of Prime Number Theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin Heidelberg New York 2000.

  44. Rhin G., Viola C.: The group structure for ζ(3). Acta Arith. XCVII, 269–293 (2001)

    Article  MathSciNet  Google Scholar 

  45. B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monat. der König. Preuss. Akad. der Wissen. zu Berlin aus dem Jahre 1859, (1860), 671–680. Also in Gesammelte Werke, Zw. Aufl. 1892; Œuvres mathématiques, Gauthiers-Villars, Paris 1898.

  46. B. Riemann, manuscript memoir (1859), available at the web address: http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/riemann1859.pdf

  47. Rivoal T.: La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris Sr. I Math. 331, 267–270 (2000)

    MATH  MathSciNet  Google Scholar 

  48. O. Schlömilch, Uebungsaufgaben für Schüler, Lehrsatz von dem Herrn Prof. Dr. Schlömilch, Archiv der Math. u. Phys. (Grunert’s Archiv), 12 (1849), 415.

  49. F. Severi, Sulle corrispondenze fra i punti di una curva algebrica e sopra certe classi di superficie, Mem. R. Accad. Sci. Torino, (2) 54 (1903). Also Opere Matematiche. Accad. Naz. Lincei, Vol. I, Roma 1971, 173–223.

  50. C.L. Siegel, Neuer Beweis für die Funktonalgleichung der Dedekindschen Zetafunktion, Math. Annalen, 85 (1922), 123–128. Also Gesammelte Abhandlungen. Bd. I, no. 7. Springer-Verlag, Berlin Heidelberg New York 1966.

  51. Terasoma T.: Mixed Tate motives and multiple zeta values. Inventiones Math. 149, 339–369 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  52. E.C. Titchmarsh, The theory of the Riemann zeta-function. Second edition. Edited and with a preface by D. R. Heath-Brown. The Clarendon Press, Oxford University Press, New York, 1986. x+412 pp.

  53. von Mangoldt H.: Zu Riemann’s Abhandlung “Ueber die Anzahl der Primzahlen unter einer gegebener Größe”. J. Reine Angew. Math 114, 255–305 (1995)

    Google Scholar 

  54. Ulmer D.: Elliptic curves with large rank over function fields. Ann. of Math., (2) 155, 295–315 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  55. H. Yoshida, On hermitian forms attached to zeta functions, in N. Kurokawa and T. Sunada (eds.) Zeta Functions in Geometry. Advances in Pure Mathematics, 21, Mathematical Society of Japan, Tokyo 1992.

  56. A.Weil, On Eisenstein’s copy of the Disquisitiones, in Algebraic Number Theory. Papers in honor of K. Iwasawa on the occasion of his 70th birthday on Sept. 11, 1987. J. Coates, R. Greenberg, B. Mazur, I. Satake ed.s, Advanced Studies in Math., 17, Academic Press, Boston MA 1989, xxvi+492 pp., in particular pp. 463–469.

  57. Weil A.: On the Riemann hypothesis in functionfields. Proc. Nat. Acad. Sci. U.S.A. 27, 345–347 (1941)

    Article  MATH  MathSciNet  Google Scholar 

  58. A. Weil, Sur les “formules explicites” de la théorie des nombres premiers, Meddelanden Från Lunds Univ. Mat. Sem. (dedié a M. Riesz) (1952), 252-265.

  59. A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent. Actualités Sci. Ind., no. 1041, Hermann et Cie., Paris, 1948. iv+85 pp.

  60. E.T. Whittaker, G.N.Watson, A Course of Modern Analysis. IVth ed., Cambridge 1927.

  61. D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992), 497–512, Progr. Math., 120, Birkhäuser, Basel, 1994. Electronic references:

  62. http://en.wikipedia.org/wiki/Leonhard_Euler also Johann_Peter_Gustav_Lejeune_Dirichlet also Bernhard_Riemann

  63. http://en.wikipedia.org/wiki/Riemann_hypothesis#Numerical_calculations

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton University, New Jersey, 08540, USA

    E. Bombieri

Authors
  1. E. Bombieri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Bombieri.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bombieri, E. The Classical Theory of Zeta and L-Functions. Milan J. Math. 78, 11–59 (2010). https://doi.org/10.1007/s00032-010-0121-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00032-010-0121-8

Mathematics Subject Classification (2010)

Search

Navigation