Computational Aspects of Hamburger’s Theorem

  • Yuri MatiyasevichEmail author
Conference paper
Part of the Fields Institute Communications book series (FIC, volume 82)


Riemann’s zeta function (defined by a certain Dirichlet series) satisfies an identity known as the functional equation. H. Hamburger established that the function is identified by the equation inside a wide class of functions defined by Dirichlet series.

Riemann’s zeta function is a member of a large family of functions with similar properties, in particular, satisfying certain functional equations. Hamburger’s theorem can be extended to some (but not to all) of these equations.

The paper addresses the following question: how could we discover the Dirichlet series satisfying given functional equation? Two “rules of thumb” for performing such discoveries via numerical computations are demonstrated for functional equations satisfied by Dirichlet eta function, Ramanujan tau L-function, and Davenport–Heilbronn function.

A conjectured discrete version of Hamburger’s theorem is stated.


  1. 1.
    Balanzario EP (2000) Remark on Dirichlet series satisfying functional equations. Divulg Mat 8(2):169–175MathSciNetzbMATHGoogle Scholar
  2. 2.
    Balanzario EP, Sánchez-Ortiz J (2007) Zeros of the Davenport–Heilbronn counterexample. Math Comput 76(260):2045–2049MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beliakov G, Matiyasevich Yu (2015) Approximation of Riemann’s zeta function by finite Dirichlet series: a multiprecision numerical approach. Exp Math 24(2):150–161. (See also Scholar
  4. 4.
    Blagouchine IV (2014) Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results. Ramanujan J 35:21–110; Addendum: Ibid, 42:777–781, 2017Google Scholar
  5. 5.
    Bohr HA, Mollerup J (1922) Lærebog i matematisk analyse af Harald Bohr og Johannes Mollerup, vol III. J. Gjellerups, Copenhagen. Google Scholar
  6. 6.
    Bombieri E, Gosh A (2011) On the Davenport–Heilbronn function. Uspekhi Mat Nauk 66:15–66. Translated in: Russian Mathematical Surveys, 66:221–270, 2011Google Scholar
  7. 7.
    Borwein P (2000) An efficient algorithm for the Riemann zeta function. In: Constructive, experimental, and nonlinear analysis (Limoges, 1999). CRC mathematical modelling series, vol 27. CRC, Boca Raton, pp 29–34Google Scholar
  8. 8.
    Euler L (1768) Remarques sur un beau rapport entre les series des puissances tant directes que reciproques. Memoires de l’Academie des sciences de Berlin 17:83–106. Reprinted in Opera omnia. Series prima: Opera mathematica. Vol. XV: Commentationes analyticae ad theoriam seriarum infinitarum pertinentes, G. Faber, ed., pp. 70–90, Leipzig, B. G. Teubner (1911,1980).
  9. 9.
    Farmer DW, Ryan NC (2014) Evaluating L-functions with few known coefficients. LMS J Comput Math 17:245–258. arXiv:1211.4181MathSciNetCrossRefGoogle Scholar
  10. 10.
    Farmer DW, Koutsoliotas S, Lemurell S (2014) Maass forms on GL(3) and GL(4). Int Math Res Not 2014(22):6276–6301MathSciNetCrossRefGoogle Scholar
  11. 11.
    Farmer DW, Koutsoliotas S, Lemurell S (2015) Varieties via their L-functions. arXive 1502.00850Google Scholar
  12. 12.
    Hamburger H (1921) Über die Riemannsche Funktionalgleichung der ζ-Funktion (Zweite Mitteilung). Math Z 11:224–245MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hamburger H (1921) Über die Riemannsche Funktionalgleichung der ζ-Funk-tion (Erste Mitteilung). Math Z 10:240–254MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kaczorowski J, Molteni G, Perelli A (2010) A converse theorem for Dirichlet L-functions. Comment Math Helv 85(2):463–483MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kinkelin H (1858) Ueber einige unendliche Reihen. Mitteilungen der Naturforschenden Gesellschaft in Bern 419–420:89–104. Google Scholar
  16. 16.
    Malmstén CJ (1849) De integralibus quibusdam definitis seriebusque infinitis. J Reine Angew Math 38:1–39MathSciNetCrossRefGoogle Scholar
  17. 17.
    Matiyasevich Yu (2012) New conjectures about zeros of Riemann’s zeta function. Research Report of the Department of Mathematics of University of Leicester, MA12-03, 44 pp.,
  18. 18.
    Matiyasevich Yu (2013) Calculation of Riemann’s zeta function via interpolating determinants. Preprint of Max Planck Institute for Mathematics in Bonn, 18, 31 pp.,
  19. 19.
    Matiyasevich Yu (2018) Computational rediscovery of Ramanujan’s tau numbers. Integers 18A:1–8. MathSciNetGoogle Scholar
  20. 20.
  21. 21.
    Matiyasevich Yu (2018) Computational aspects of Hamburger’s theorem. Preprint POMI 18-01, 31pp.
  22. 22.
    Perelli A (2017) Converse theorems: from the Riemann zeta function to the Selberg class. Bollettino dell’Unione Matematica Italiana 10(1):29–53. (see also ArXiv 1605.02354)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Riemann B (1859) Über die Anzhal der Primzahlen unter einer gegebenen Grösse. Monatsberichter der Berliner Akademie. Included into: Riemann, B. Gesammelte Werke. Teubner, Leipzig, 1892; reprinted by Dover Books, New York, 1953., English translation:
  24. 24.
    Rubinstein M (2005) Computational methods and experiments in analytic number theory. In: Recent perspectives in random matrix theory and number theory. Proceedings of a school that was part of the programme ‘Random matrix approaches in number theory’, Cambridge, UK, January 26–July 16, 2004. Cambridge University Press, Cambridge, pp 425–506Google Scholar
  25. 25.
    Selberg A (1992) Old and new conjectures and results about a class of Dirichlet series. In Proceedings of the Amalfi conference on analytic number theory (Maiori, 1989). Univ. Salerno, Salerno, pp 367–385. Reprinted in Collected papers, Vol. II, Springer-Verlag, 1991 and 2014Google Scholar
  26. 26.
    Sloane NJA (ed) The on-line encyclopedia of integer sequences.
  27. 27.
    Titchmarsh EC (1986) The theory of the Riemann zeta-function, 2nd edn. The Clarendon Press, New YorkzbMATHGoogle Scholar
  28. 28.
    Turing AM (1953) Some calculations of the Riemann zeta-function. Proc Lond Math Soc 3:99–117. Reprinted in: Collected Works of A. M. Turing: Pure Mathematics (J. L. Britton, ed.), North-Holland, Amsterdam, (1992); Alan Turing – His Work and Impact, S. B. Cooper, J. van Leeuwen, eds., Elsevier Science, 2013. ISBN: 978-0-12-386980-7Google Scholar
  29. 29.
    Wilton JR (1927) A note on Ramanujan’s arithmetical function τ(n). Math Proc Camb Philos Soc 23(6):675–680CrossRefGoogle Scholar
  30. 30.
    The inverse symbolic calculator.
  31. 31.

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St. Petersburg Department of V. A. Steklov Mathematical InstituteSaint PetersburgRussia

Personalised recommendations