Journal of the Mathematical Society of Japan

Analytic continuation of multiple Hurwitz zeta functions

Jay MEHTA and G. K. VISWANADHAM

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Abstract

We obtain the analytic continuation of multiple Hurwitz zeta functions by using a simple and elementary translation formula. We also locate the polar hyperplanes for these functions and express the residues, along these hyperplanes, as coefficients of certain infinite matrices.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 4 (2017), 1431-1442.

Dates
First available in Project Euclid: 25 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1508918563

Digital Object Identifier
doi:10.2969/jmsj/06941431

Mathematical Reviews number (MathSciNet)
MR3715810

Zentralblatt MATH identifier
06821646

Subjects
Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values 11M35: Hurwitz and Lerch zeta functions

Keywords
multiple Hurwitz zeta functions analytic continuation

Citation

MEHTA, Jay; VISWANADHAM, G. K. Analytic continuation of multiple Hurwitz zeta functions. J. Math. Soc. Japan 69 (2017), no. 4, 1431--1442. doi:10.2969/jmsj/06941431. https://projecteuclid.org/euclid.jmsj/1508918563


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References

  • S. Akiyama, S. Egami and Y. Tanigawa, Analytic continuation of multiple zeta functions and their values at non-positive integers, Acta Arith., 98 (2001), 107–116.
  • S. Akiyama and H. Ishikawa, On analytic continuation of multiple $L$-functions and related zeta functions, Analytic number theory (Beijing/Kyoto, 1999) Dev. Math., 6 (2002), 1–16.
  • S. Gun and B. Saha, Multiple Lerch zeta functions and an idea of Ramanujan, to appear in Michigan Math. J., arXiv:1510.05835.
  • J. Ecalle, ARI/GARI, et la décomposition des multizêtas en irréductibles, preprint, 2000.
  • J. Ecalle, ARI/GARI, la dimorphie et l'arithmétique des multizêtas: un premier bilan, J. Théor. Nombres Bordeaux, 15 (2003), 411–478.
  • D. Essouabri, Zeta functions associated to Pascal's triangle$\mod p$, Japan. J. Math., 31 (2005), 157–174.
  • K. Inkeri, The real roots of Bernoulli polynomials, Ann. Univ. Turk. Ser. A, 37 (1959), 20pp.
  • J. P. Kelliher and R. Masri, Analytic continuation of multiple Hurwitz zeta functions, Math. Proc. Cambridge Philos. Soc., 145 (2008), 605–617.
  • Y. Komori, An integral representation of multiple Hurwitz–Lerch zeta functions and generalized multiple Bernoulli numbers, Q. J. Math., 61 (2010), 437–496.
  • K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math. J., 172 (2003), 59–102.
  • J. Mehta, B. Saha and G. K. Viswanadham, Analytic properties of multiple zeta functions and certain weighted variants, an elementary approach, J. Number Theory, 168 (2016), 487–508.
  • M. R. Murty and K. Sinha, Multiple Hurwitz zeta functions. Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Sympos. Pure Math., 75 (2006), 135–156.
  • S. Ramanujan, A series for Euler's constant $\gamma$, Messenger of Mathematics, 46 (1917), 73–80.
  • B. Saha, On the analytic continuation of multiple Dirichlet series and their singularities, Ph. D. Thesis, HBNI, 2016.