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Algorithms and Complexity in Mathematics, Epistemology, and Science pp 195–223Cite as

Computational Aspects of Hamburger’s Theorem

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Part of the book series: Fields Institute Communications ((FIC,volume 82))

Abstract

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.

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Notes

  1. 1.

    In this respect it is interesting to note that A. Turing in [28] used the word “computer” having in mind “a man performing computations”; in this sense a computer (namely, Euler) was involved in the discovery of (6).

  2. 2.

    This form of writing the functional identity is due to Kinkelin [15], Euler and Malmstén worked with an equivalent formula in terms of function η(s); Riemann mentioned neither Euler nor Malmstén (for other historical details see, for example, [4]).

  3. 3.

    The author is grateful to the referee for indicating to these papers of which the author was ignorant at the time of writing [21].

  4. 4.

    This example of a pair of functions solving the same functional equation was considered by E. P. Balanzario and J. Sánchez-Ortiz in [1, 2].

References

  1. Balanzario EP (2000) Remark on Dirichlet series satisfying functional equations. Divulg Mat 8(2):169–175

    MathSciNet  MATH  Google Scholar 

  2. Balanzario EP, Sánchez-Ortiz J (2007) Zeros of the Davenport–Heilbronn counterexample. Math Comput 76(260):2045–2049

    Article  MathSciNet  Google Scholar 

  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 http://arxiv.org/abs/1402.5295)

    Article  MathSciNet  Google Scholar 

  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, 2017

    Google Scholar 

  5. Bohr HA, Mollerup J (1922) Lærebog i matematisk analyse af Harald Bohr og Johannes Mollerup, vol III. J. Gjellerups, Copenhagen. https://books.google.ru/books?id=RIpVAAAAYAAJ

    Google Scholar 

  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, 2011

    Google Scholar 

  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 29–34

    Google Scholar 

  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). http://eulerarchive.maa.org/pages/E352.html

  9. Farmer DW, Ryan NC (2014) Evaluating L-functions with few known coefficients. LMS J Comput Math 17:245–258. arXiv:1211.4181

    Article  MathSciNet  Google Scholar 

  10. Farmer DW, Koutsoliotas S, Lemurell S (2014) Maass forms on GL(3) and GL(4). Int Math Res Not 2014(22):6276–6301

    Article  MathSciNet  Google Scholar 

  11. Farmer DW, Koutsoliotas S, Lemurell S (2015) Varieties via their L-functions. arXive 1502.00850

    Google Scholar 

  12. Hamburger H (1921) Über die Riemannsche Funktionalgleichung der ζ-Funktion (Zweite Mitteilung). Math Z 11:224–245

    Article  MathSciNet  Google Scholar 

  13. Hamburger H (1921) Über die Riemannsche Funktionalgleichung der ζ-Funk-tion (Erste Mitteilung). Math Z 10:240–254

    Article  MathSciNet  Google Scholar 

  14. Kaczorowski J, Molteni G, Perelli A (2010) A converse theorem for Dirichlet L-functions. Comment Math Helv 85(2):463–483

    Article  MathSciNet  Google Scholar 

  15. Kinkelin H (1858) Ueber einige unendliche Reihen. Mitteilungen der Naturforschenden Gesellschaft in Bern 419–420:89–104. https://books.google.de/books?id=N3cWAQAAIAAJ&pg=PA89#v=onepage&q&f=false

    Google Scholar 

  16. Malmstén CJ (1849) De integralibus quibusdam definitis seriebusque infinitis. J Reine Angew Math 38:1–39

    Article  MathSciNet  Google Scholar 

  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. http://www2.le.ac.uk/departments/mathematics/research/research-reports-2/reports-2012/ma12-03, https://logic.pdmi.ras.ru/~yumat/talks/leicester_2012/MA12_03Matiyasevich.pdf

  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. http://www.mpim-bonn.mpg.de/preblob/5368, https://logic.pdmi.ras.ru/~yumat/talks/bonn_2013/5368.pdf

  19. Matiyasevich Yu (2018) Computational rediscovery of Ramanujan’s tau numbers. Integers 18A:1–8. http://math.colgate.edu/~integers/vol18a.html

    MathSciNet  Google Scholar 

  20. Matiyasevich Yu WWW personal journal. https://logic.pdmi.ras.ru/~yumat/personaljournal/finitedirichlet

  21. Matiyasevich Yu (2018) Computational aspects of Hamburger’s theorem. Preprint POMI 18-01, 31pp. http://www.pdmi.ras.ru/preprint/2018/18-01.html

  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)

    Article  MathSciNet  Google Scholar 

  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. http://www.claymath.org/publications/riemanns-1859-manuscript, English translation: http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf

  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 425–506

    Google Scholar 

  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 2014

    Google Scholar 

  26. Sloane NJA (ed) The on-line encyclopedia of integer sequences. https://oeis.org/A000594

  27. Titchmarsh EC (1986) The theory of the Riemann zeta-function, 2nd edn. The Clarendon Press, New York

    MATH  Google Scholar 

  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: 978-0-12-386980-7

    Google Scholar 

  29. Wilton JR (1927) A note on Ramanujan’s arithmetical function τ(n). Math Proc Camb Philos Soc 23(6):675–680

    Article  Google Scholar 

  30. The inverse symbolic calculator. https://isc.carma.newcastle.edu.au/standard

  31. Wolframalpha. https://www.wolframalpha.com

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Authors and Affiliations

  1. St. Petersburg Department of V. A. Steklov Mathematical Institute, Saint Petersburg, Russia

    Yuri Matiyasevich

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  1. Yuri Matiyasevich
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Correspondence to Yuri Matiyasevich .

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Editors and Affiliations

  1. Department of Philosophy, Simon Fraser University Philosophy, Burnaby, BC, Canada

    Nicolas Fillion

  2. The Rotman Institute of Philosophy, The Ontario Research Center for Computer Algebra and The School of Mathematical and Statistical Sciences, The University of Western Ontario, London, ON, Canada

    Robert M. Corless

  3. Department of Physics & Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias S. Kotsireas

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Matiyasevich, Y. (2019). Computational Aspects of Hamburger’s Theorem. In: Fillion, N., Corless, R., Kotsireas, I. (eds) Algorithms and Complexity in Mathematics, Epistemology, and Science. Fields Institute Communications, vol 82. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9051-1_8

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