Journal of the Mathematical Society of Japan

Mean value theorems for the double zeta-function

Kohji MATSUMOTO and Hirofumi TSUMURA

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We prove asymptotic formulas for mean square values of the Euler double zeta-function $\zeta_2(s_0,s)$, with respect to $\Im s$. Those formulas enable us to propose a double analogue of the Lindelöf hypothesis.

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J. Math. Soc. Japan, Volume 67, Number 1 (2015), 383-406.

First available in Project Euclid: 22 January 2015

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Zentralblatt MATH identifier

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$

double zeta-functions mean values Lindelöf hypothesis Euler's constant


MATSUMOTO, Kohji; TSUMURA, Hirofumi. Mean value theorems for the double zeta-function. J. Math. Soc. Japan 67 (2015), no. 1, 383--406. doi:10.2969/jmsj/06710383.

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  • 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.
  • T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J., 153 (1999), 189–209.
  • F. V. Atkinson, The mean-value of the Riemann zeta function, Acta Math., 81 (1949), 353–376.
  • D. Borwein, J. M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc. (2), 38 (1995), 277–294.
  • H. Ishikawa and K. Matsumoto, On the estimation of the order of Euler-Zagier multiple zeta-functions, Illinois J. Math., 47 (2003), 1151–1166.
  • I. Kiuchi and Y. Tanigawa, Bounds for double zeta-functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5 (2006), 445–464.
  • I. Kiuchi, Y. Tanigawa and W. Zhai, Analytic properties of double zeta-functions, Indag. Math. (N.S.), 21 (2011), 16–29.
  • Y. Komori, K. Matsumoto and H. Tsumura, Functional equations and functional relations for the Euler double zeta-function and its generalization of Eisenstein type, Publ. Math. Debrecen, 77 (2010), 15–31.
  • K. Matsumoto, On the analytic continuation of various multiple zeta-functions, In: Number Theory for the Millennium. II, Urbana, IL, 2000, (eds. M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand and W. Philipp), A K Peters, Natick, MA, 2002, pp.,417–440.
  • K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions. I, J. Number Theory, 101 (2003), 223–243.
  • K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math. J., 172 (2003), 59–102.
  • K. Matsumoto, Functional equations for double zeta-functions, Math. Proc. Cambridge Philos. Soc., 136 (2004), 1–7.
  • K. Matsumoto and H. Tsumura, Mean value theorems for double zeta-functions, In: Analytic Number Theory–-Number Theory through Approximation and Asymptotics, RIMS, 2012, (ed. K. Chinen), RIMS Kôkyûroku, 1874, RIMS, 2014, pp.,45–54.
  • T. Nakamura and \L. Pańkowski, Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line, preprint, arXiv:math.NT/1212.5890.
  • E. C. Titchmarsh, The Theory of the Riemann Zeta-function. 2nd ed., Edited and with a preface by D. R. Heath-Brown, The Clarendon Press, Oxford University Press, New York, 1986.