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On certain mean values of the double zeta-function

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  11 January 2016

Soichi Ikeda ,
Kaneaki Matsuoka  and
Yoshikazu Nagata
Soichi Ikeda
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan, m10004u@math.nagoya-u.ac.jp
Kaneaki Matsuoka
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku Nagoya 464-8602, Japan, m10041v@math.nagoya-u.ac.jp
Yoshikazu Nagata
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku Nagoya 464-8602, Japan, m10035y@math.nagoya-u.ac.jp
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Abstract

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In this article we discuss three types of mean values of the Euler double zeta-function. To get the results, we introduce three approximate formulas for this function.

MSC classification

Secondary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values 11M06: $zeta (s)$ and $L(s, chi)$
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 217 , March 2015 , pp. 161 - 190
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Akiyama, S., Egami, S., and Tanigawa, Y., Analytic continuation of multiple zeta- functions and their values at non-positive integers, Acta Arith. 98 (2001), 107–116. MR 1831604. DOI 10.4064/aa98-2-1.Google Scholar
[2] Atkinson, F. V., The mean-value of the Riemann zeta function, Acta Math. 81 (1949), 353–376. MR 0031963.Google Scholar
[3] Edwards, H. M., Riemann's Zeta Function, Pure Appl. Math. 58, Academic Press, New York, 1974. MR 0466039.Google Scholar
[4] Hoffman, M. E., Multiple harmonic series, Pacific J. Math. 152 (1992), 275–290. MR 1141796.Google Scholar
[5] Ivić, A., The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications, Wiley-Intersci. Publ., Wiley, New York, 1985. MR 0792089.Google Scholar
[6] Kiuchi, I. and Tanigawa, Y., Bounds for double zeta-functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), 445–464. MR 2297719.Google Scholar
[7] Kiuchi, I., Tanigawa, Y., and Zhai, W., Analytic properties of double zeta-functions, Indag. Math. (N.S.) 21 (2011), 16–29. MR 2832479. DOI 10.1016/j.indag.2010.12.001.CrossRefGoogle Scholar
[8] Matsumoto, K., “On the analytic continuation of various multiple zeta-functions” in Number Theory for the Millennium, II (Urbana, 2000), A. K. Peters, Natick, Mass., 2002, 417–440. MR 1956262.Google Scholar
[9] Matsumoto, K., Functional equations for double zeta-functions, Math. Proc. Cambridge Phi los. Soc. 136 (2004), 1–7. MR 2034011. DOI 10.1017/S0305004103007035.CrossRefGoogle Scholar
[10] Matsumoto, K. and Tsumura, H., Mean value theorems for the double zeta-function, J. Math. Soc. Japan 67 (2015), 383–406.MR 3304026.CrossRefGoogle Scholar
[11] Shimomura, S., Fourth moment of the Riemannzeta-functionwithashift alongthe real line, Tokyo J. Math. 36 (2013), 355–377. MR 3161563. DOI 10.3836/tjm/1391177976.Google Scholar
[12] Titchmarsh, E. C., The Theory of Functions, reprint of the 2nd ed., Oxford University Press, Oxford, 1958. MR 3155290.Google Scholar
[13] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, 2nd ed., Oxford University Press, New York, 1986. MR 0882550.Google Scholar
[14] Zagier, D., “Values of zeta-functions and their applications” in First European Congress of Mathematics,Vol. II (Paris, 1992) , Progr. Math. 120, Birkhäuser, Basel, 1994, 497–512. MR 1341859.Google Scholar
[15] Zhao, J. Q., Analytic continuation of multiple zeta function, Proc. Amer. Math. Soc. 128 (2000), 1275–1283. MR 1670846. DOI 10.1090/S0002-9939-99-05398-8.Google Scholar