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On functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function

Published online by Cambridge University Press:  01 May 2007

Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo192-0397, Japan.


In this paper, we give certain analytic functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell–Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950's.

Research Article
Copyright © Cambridge Philosophical Society 2007

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[1]Erdelyi, A. (Dir.), Magnus, W., Oberthettinger, F. and Tricomi, F.. Higher Transcendental Functions. Vol. 1 (McGraw-Hill, 1953).Google Scholar
[2]Huard, J. G., Williams, K. S. and Zhang, Nan-Yue. On Tornheim's double series.Acta Arith. 75 (1996), 105117.Google Scholar
[3]Matsumoto, K.. On the analytic continuation of various multiple zeta-functions. Number Theory for the Millennium II, Proc. of the Millennial Conference on Number Theory,Bennett, M. A. et al. . (eds.) (A. K. Peters, 2002), 417440.Google Scholar
[4]Matsumoto, K.. On Mordell–Tornheim and other multiple zeta-functions. Proceedings of the Session in analytic number theory and Diophantine equations Bonn, January–June 2002), Heath-Brown, D. R. and Moroz, B. Z. eds.). Bonner Mathematische Schriften Nr. 360 (Bonn, 2003), n. 25, 17 pp.Google Scholar
[5]Matsumoto, K.. Functional equations for double zeta-functions. Math. Proc. Camb. Phil. Soc. 136 (2004), 17.CrossRefGoogle Scholar
[6]Mordell, L. J.. On the evaluation of some multiple series. J. London Math. Soc. 33 (1958), 368371.CrossRefGoogle Scholar
[7]Subbarao, M. V. and Sitaramachandrarao, R.. On some infinite series of L. J. Mordell and their analogues. Pacific J. Math. 119 (1985), 245255.CrossRefGoogle Scholar
[8]Tornheim, L.. Harmonic double series. Amer. J. Math. 72 (1950), 303314.Google Scholar
[9]Tsumura, H.. On some combinatorial relations for Tornheim's double series. Acta Arith. 105 (2002), 239252.Google Scholar
[10]Tsumura, H.. On alternating analogues of Tornheim's double series. Proc. Amer. Math. Soc. 131 (2003), 36333641.Google Scholar
[11]Tsumura, H.. Certain functional relations for the double harmonic series related to the double Euler numbers. J. Austral. Math. Soc., Ser. A. 79 (2005), 319333.CrossRefGoogle Scholar
[12]Zagier, D.. Values of zeta functions and their applications. Proc. First Congress of Math. Paris, vol. II Birkhäuser Verlag, 1994), 497512.Google Scholar