## Functional relations and special values of Mordell-Tornheim triple zeta and $L$-functions

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- by Kohji Matsumoto, Takashi Nakamura and Hirofumi Tsumura PDF
- Proc. Amer. Math. Soc.
**136**(2008), 2135-2145 Request permission

## Abstract:

In this paper, we prove the existence of meromorphic continuation of certain triple zeta-functions of Lerch’s type. Based on this result, we prove some functional relations for triple zeta and $L$-functions of the Mordell-Tornheim type. Using these functional relations, we prove new explicit evaluation formulas for special values of these functions. These can be regarded as triple analogues of known results for double zeta and $L$-functions.## References

- Tom M. Apostol,
*Introduction to analytic number theory*, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR**0434929** - Tsuneo Arakawa and Masanobu Kaneko,
*On multiple $L$-values*, J. Math. Soc. Japan**56**(2004), no. 4, 967–991. MR**2091412**, DOI 10.2969/jmsj/1190905444 - Kohji Matsumoto,
*On the analytic continuation of various multiple zeta-functions*, Number theory for the millennium, II (Urbana, IL, 2000) A K Peters, Natick, MA, 2002, pp. 417–440. MR**1956262** - D. R. Heath-Brown and B. Z. Moroz (eds.),
*Proceedings of the Session in Analytic Number Theory and Diophantine Equations*, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 360, Universität Bonn, Mathematisches Institut, Bonn, 2003. Held in Bonn, January–June, 2002. MR**2072675** - K. Matsumoto, T. Nakamura, H. Ochiai and H. Tsumura,
*On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions*, to appear in Acta Arith. - L. J. Mordell,
*On the evaluation of some multiple series*, J. London Math. Soc.**33**(1958), 368–371. MR**100181**, DOI 10.1112/jlms/s1-33.3.368 - Takashi Nakamura,
*A functional relation for the Tornheim double zeta function*, Acta Arith.**125**(2006), no. 3, 257–263. MR**2276193**, DOI 10.4064/aa125-3-3 - T. Nakamura,
*Double Lerch series and their functional relations*, to appear in Aequationes Math. - David Terhune,
*Evaluation of double $L$-values*, J. Number Theory**105**(2004), no. 2, 275–301. MR**2040159**, DOI 10.1016/j.jnt.2003.11.007 - David Terhune,
*Evaluations of a class of double $L$-values*, Proc. Amer. Math. Soc.**134**(2006), no. 7, 1881–1889. MR**2215115**, DOI 10.1090/S0002-9939-05-08261-4 - Leonard Tornheim,
*Harmonic double series*, Amer. J. Math.**72**(1950), 303–314. MR**34860**, DOI 10.2307/2372034 - Hirofumi Tsumura,
*On some combinatorial relations for Tornheim’s double series*, Acta Arith.**105**(2002), no. 3, 239–252. MR**1931792**, DOI 10.4064/aa105-3-3 - Hirofumi Tsumura,
*On alternating analogues of Tornheim’s double series*, Proc. Amer. Math. Soc.**131**(2003), no. 12, 3633–3641. MR**1998168**, DOI 10.1090/S0002-9939-03-07186-7 - Hirofumi Tsumura,
*Evaluation formulas for Tornheim’s type of alternating double series*, Math. Comp.**73**(2004), no. 245, 251–258. MR**2034120**, DOI 10.1090/S0025-5718-03-01572-2 - Hirofumi Tsumura,
*On evaluation formulas for double $L$-values*, Bull. Austral. Math. Soc.**70**(2004), no. 2, 213–221. MR**2094289**, DOI 10.1017/S0004972700034432 - Hirofumi Tsumura,
*On Mordell-Tornheim zeta values*, Proc. Amer. Math. Soc.**133**(2005), no. 8, 2387–2393. MR**2138881**, DOI 10.1090/S0002-9939-05-08132-3 - Hirofumi Tsumura,
*On some functional relations between Mordell-Tornheim double $L$-functions and Dirichlet $L$-functions*, J. Number Theory**120**(2006), no. 1, 161–178. MR**2256802**, DOI 10.1016/j.jnt.2005.11.006 - Hirofumi Tsumura,
*On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function*, Math. Proc. Cambridge Philos. Soc.**142**(2007), no. 3, 395–405. MR**2329691**, DOI 10.1017/S0305004107000059 - Lawrence C. Washington,
*Introduction to cyclotomic fields*, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR**1421575**, DOI 10.1007/978-1-4612-1934-7 - Don Zagier,
*Values of zeta functions and their applications*, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 497–512. MR**1341859**

## Additional Information

**Kohji Matsumoto**- Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
- Email: kohjimat@math.nagoya-u.ac.jp
**Takashi Nakamura**- Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
- MR Author ID: 755913
- Email: m03024z@math.nagoya-u.ac.jp
**Hirofumi Tsumura**- Affiliation: Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan
- Email: tsumura@tmu.ac.jp
- Received by editor(s): August 31, 2006
- Received by editor(s) in revised form: April 11, 2007
- Published electronically: February 21, 2008
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**136**(2008), 2135-2145 - MSC (2000): Primary 40B05; Secondary 11M35, 11M06, 33E20
- DOI: https://doi.org/10.1090/S0002-9939-08-09192-2
- MathSciNet review: 2383519