Skip to main content
SpringerLink
Log in

Identities for the Hurwitz zeta function, Gamma function, and L-functions

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet L-functions. They involve a sequence of polynomials α k (s) whose study was initiated in Rubinstein (Ramanujan J. 27(1): 29–42, 2012). The expansions given here are practical and can be used for the high precision evaluation of these functions, and for deriving formulas for special values. We also present a summation formula and use it to generalize a formula of Hasse.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amore, P.: Convergence acceleration of series through a variational approach. J. Math. Anal. Appl. 323(1), 63–77 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cohen, H.: Number Theory, Volume II: Analytic and Modern Tools. Graduate Texts in Mathematics, vol. 240. Springer, Berlin (2007)

    Google Scholar 

  3. Coffey, M.W.: Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant. Eprint. arXiv:1106.5146

  4. Davenport, H.: Multiplicative Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 74. Springer, Berlin (1980)

    Book  MATH  Google Scholar 

  5. Hasse, H.: Ein Summierungsverfahren für die Riemannsche Zeta-Reihe. Math. Z. 32, 458–464 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kenter, F.K.: A matrix representation for Euler’s constant, γ. Am. Math. Mon. 106(5), 452–454 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kluyver, J.C.: Euler’s constant and natural numbers. Proc. K. Ned. Akad. Wet. 27(1–2), 142–144 (1924)

    Google Scholar 

  8. Rubinstein, M.O.: Identities for the Riemann zeta function. Ramanujan J. 27(1), 29–42 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rubinstein, M.O.: Computational methods and experiments in analytic number theory. In: Mezzadri, F., Snaith, N.C. (eds.) Recent Perspectives in Random Matrix Theory and Number Theory. London Mathematical Society Lecture Note Series, vol. 322, pp. 425–506. Cambridge University Press, Cambridge (2005)

    Chapter  Google Scholar 

  10. Sondow, J.: Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series. Proc. Am. Math. Soc. 120, 421–424 (1994)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pure Mathematics, University of Waterloo, 200 University Ave W, Waterloo, ON, N2L 3G1, Canada

    Michael O. Rubinstein

Authors
  1. Michael O. Rubinstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael O. Rubinstein.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rubinstein, M.O. Identities for the Hurwitz zeta function, Gamma function, and L-functions. Ramanujan J 32, 421–464 (2013). https://doi.org/10.1007/s11139-013-9468-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-013-9468-0

Keywords

Mathematics Subject Classification

Search

Navigation