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Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series

Published online by Cambridge University Press:  22 January 2016

Kohji Matsumoto
Kohji Matsumoto*
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, [email protected]
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Abstract

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The present paper contains three main results. The first is asymptotic expansions of Barnes double zeta-functions, and as a corollary, asymptotic expansions of holomorphic Eisenstein series follow. The second is asymptotic expansions of Shintani double zeta-functions, and the third is the analytic continuation of n-variable multiple zeta-functions (or generalized Euler-Zagier sums). The basic technique of proving those results is the method of using the Mellin-Barnes type of integrals.

Keywords

11M41
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 172 , 2003 , pp. 59 - 102
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

References

References

[1] Akiyama, S., Egami, S. and Tanigawa, Y., An analytic continuation of multiple zeta functions and their values at non-positive integers, Acta Arith., 98 (2001), 107–116.CrossRefGoogle Scholar
[2] Akiyama, S. and Ishikawa, H., On analytic continuation of multiple L-functions and related zeta-functions, Analytic Number Theory (Jia, C. and Matsumoto, K., eds.), Developments in Math. Vol. 6, Kluwer Acad. Publishers (2002), pp. 1–16.Google Scholar
[3] Arakawa, T. and Kaneko, M., Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J., 153 (1999), 189–209.Google Scholar
[4] Atkinson, F. V., The mean-value of the Riemann zeta function, Acta Math., 81 (1949), 353–376.Google Scholar
[5] Barnes, E. W., The theory of the double gamma function, Philos. Trans. Roy. Soc.(A), 196 (1901), 265–387.Google Scholar
[6] Barnes, E. W., On the theory of multiple gamma function, Trans. Cambridge Phil. Soc, 19 (1904), 374–425.Google Scholar
[7] Katsurada, M., An application of Mellin-Barnes’ type integrals to the mean square of Lerch zeta-functions, Collect. Math., 48 (1997), 137–153.Google Scholar
[8] Katsurada, M., An application of Mellin-Barnes type of integrals to the mean square of L-functions, Liet. Mat. Rink. 38 (1998), 98–112. = Lithuanian Math. J. 38 (1998), 77–88.Google Scholar
[9] Katsurada, M., Power series and asymptotic series associated with the Lerch zeta-function, Proc. Japan Acad. Ser. A, 74 (1998), 167–170.CrossRefGoogle Scholar
[10] Katsurada, M. and Matsumoto, K., Asymptotic expansions of the mean values of Dirichlet L-functions, Math. Z., 208 (1991), 23–39.Google Scholar
[11] Katsurada, M. and Matsumoto, K., Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions I, Math. Scand., 78 (1996), 161–177.Google Scholar
[12] Lewittes, J., Analytic continuation of the series ∑(m + nz)−s , Trans. Amer. Math. Soc, 159 (1971), 505–509.Google Scholar
[13] Lewittes, J., Analytic continuation of Eisenstein series, ibid., 171 (1972), 469–490.Google Scholar
[14] Matsumoto, K., Asymptotic series for double zeta, double gamma, and Hecke L-functions, Math. Proc. Cambridge Phil. Soc, 123 (1998), 385–405.CrossRefGoogle Scholar
[15] Matsumoto, K., Corrigendum and addendum to “Asymptotic series for double zeta, double gamma, and Hecke L-functions”, ibid., 132 (2002), 377–384.Google Scholar
[16] Mikolás, M., Mellinsche Transformation und Orthogonalität bei ζ(s,u); Verallge-meinerung der Riemannschen Funktionalgleichung von ζ(s), Acta Sci. Math. Szeged, 17 (1956), 143–164.Google Scholar
[17] Shintani, T., On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976), 393–417.Google Scholar
[18] Shintani, T., On a Kronecker limit formula for real quadratic fields, ibid., 24 (1977), 167–199.Google Scholar
[19] Shintani, T., On values at s = 1 of certain L functions of totally real algebraic number fields, Algebraic Number Theory (Iyanaga, S., ed.), Japan Soc. Promot. Sci., Tokyo (1977), pp. 201–212.Google Scholar
[20] Shintani, T., A proof of the classical Kronecker limit formula, Tokyo J. Math., 3 (1980), 191–199.Google Scholar
[21] Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford, 1951.Google Scholar
[22] Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, 1927.Google Scholar
[23] Zagier, D., Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II, Invited Lectures (Part 2) (Joseph, A. et al., eds.), Progress in Math. Vol.120, Birkhäuser (1994), pp. 497–512.Google Scholar
[24] Zhao, J., Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc., 128 (2000), 1275–1283.Google Scholar

References added on August 2002

[25] Egami, S. and Matsumoto, K., Asymptotic expansions of multiple zeta-functions and power mean values of Hurwitz zeta-functions, J. London Math. Soc., (2) 66 (2002), 41–60.CrossRefGoogle Scholar
[26] Matsumoto, K., Asymptotic expansions of double gamma-functions and related remarks, Analytic Number Theory (Jia, C. and Matsumoto, K., eds.), Developments in Math. Vol. 6, Kluwer Acad. Publishers (2002), pp. 243–268.Google Scholar
[27] Matsumoto, K., On the analytic continuation of various multiple zeta-functions, Number Theory for the Millennium II, Proc. Millennial Conf. on Number Theory (Bennett, M. A. et al., eds.), A K Peters (2002), pp. 417–440.Google Scholar
[28] Matsumoto, K., Functional equations for double zeta-functions, to appear, Math. Proc. Cambridge Phil. Soc. Google Scholar
[29] Matsumoto, K., The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I, J. Number Theory, 101 (2003), 223–243.CrossRefGoogle Scholar
[30] Matsumoto, K., The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions II, Analytic and Probabilistic Methods in Number Theory, Proc. 3rd Intern. Conf. in Honour of J. Kubilius (Dubickas, A. et al., eds.) (2002), pp. 188–194.Google Scholar
[31] Matsumoto, K., On Mordell-Tornheim and other multiple zeta-functions, Proc. Bonn Workshop on Analytic Number Theory, to appear.Google Scholar
[32] Matsumoto, K. and Tanigawa, Y., The analytic continuation and the order estimate of multiple Dirichlet series, 15, J. Théorie des Nombres de Bordeaux, pp. 267–274.Google Scholar
[33] Matsumoto, K. and Weng, L., Zeta-functions defined by two polynomials, Number Theoretic Methods — Future Trends (Kanemitsu, S. and Jia, C., eds.), Developments in Math. Vol. 8, Kluwer Acad. Publishers (2002), pp. 233–262.Google Scholar