## Tokyo Journal of Mathematics

### Double Lerch Value Relations and Functional Relations for Witten Zeta Functions

Takashi NAKAMURA

#### Abstract

In this paper, we obtain functional relations for Witten zeta functions by using relations of double Lerch values. By these functional relations, we obtain new proofs of known results on the Tornheim double zeta values, the Euler-Zagier double zeta values, and their alternating and character analogues.

#### Article information

Source
Tokyo J. of Math. Volume 31, Number 2 (2008), 551-574.

Dates
First available in Project Euclid: 5 February 2009

https://projecteuclid.org/euclid.tjm/1233844070

Digital Object Identifier
doi:10.3836/tjm/1233844070

Mathematical Reviews number (MathSciNet)
MR2477890

Zentralblatt MATH identifier
1181.11056

#### Citation

NAKAMURA, Takashi. Double Lerch Value Relations and Functional Relations for Witten Zeta Functions. Tokyo J. of Math. 31 (2008), no. 2, 551--574. doi:10.3836/tjm/1233844070. https://projecteuclid.org/euclid.tjm/1233844070

#### References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.
• T. Arakawa and M. Kaneko, On multiple $L$-values, J. Math. Soc. Japan, 56 (2004), no. 4, 967–991.
• E. Gunnells and R. Sczech, Evaluation of Dedekind sums, Eisenstein cocycles, and special values of $L$-functions., Duke Math. J. 118 (2003), no. 2, 229–260.
• J. G. Huard, K. S. Williams and Z. Y. Zhang, On Tornheim's double series, Acta Arith., 75 (1996), 239–252.
• Y. Komori, K. Matsumoto and H. Tsumura, Zeta-functions of root systems, to appear in, Conference on L-functions, L. Weng and M. Kaneko (eds.), World Scientific, 2007.
• Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras III, preprint.
• K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, in, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, 17 pp., Bonner Math. Schriften, 360, Univ. Bonn, Bonn, 2003.
• K. Matsumoto, T. Nakamura, H. Ochiai and H. Tsumura, On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions, preprint.
• K. Matsumoto, T. Nakamura and H. Tsumura, Functional relations and special values of Mordell-Tornheim triple zeta-functions and L-functions, to appear in Proc. Amer. Math., Soc..
• K. Matsumoto and H. Tsumura, A new method of producing functional relations among multiple zeta functions, to appear in Quart. J., Math.
• K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras I, Ann. Inst. Fourier, 56 (2006), no. 5, 1457–1504.
• L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc., 33 (1958), 368–371.
• T. Nakamura, A functional relations for the Tornheim double zeta function, Acta Arith, 125 (2006), 257–263.
• T. Nakamura, Double Lerch series and their functional relations, to appear in Aequationes, Mathematicae.
• T. Nakamura, Functional relations related to Witten zeta functions, preprint.
• D. Terhune, Evaluation of double $L$-values, J. Number Theory, 105 (2004), no. 2, 275–301.
• L. Tornheim, Harmonic double series, Amer. J. Math., 72 (1950), 303–314.
• H. Tsumura, On alternating analogues of Tornheim's double series, Proc. Amer. Math. Soc., 131 (2003), no. 12, 3633–3641.
• H. Tsumura, Evaluation formulae for Tornheim's type of alternating double series, Math. Comp., 73 (2004), no. 245, 251–258.
• H. Tsumura, On Witten's type of zeta values attached to $\rm SO(5)$, Arch. Math. (Basel), 82 (2004), no. 2, 147–152.
• H. Tsumura, On evaluation formulae for double $L$-values, Bull. Austral. Math. Soc., 70 (2004), no. 2, 213–221.
• H. Tsumura, Certain functional relations for the double harmonic series related to the double Euler numbers, J. Aust. Math. Soc., 79 (2005), no. 3, 319–333.
• H. Tsumura, On some functional relations between Mordell-Tornheim double $L$-functions and Dirichlet $L$-functions, J. Number Theory, 120 (2006), 161–178.
• H. Tsumura, On alternating analogues of Tornheim's double series II, preprint.
• H. Tsumura, On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function," Math. Proc. Camb. Phil. Soc., 142, (2007) 395–405.
• H. Tsumura, Certain functional relations for double $L$-functions, preprint.,
• E. Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys., 141 (1991), no. 1, 153–209.
• Maoxiang Wu, On analytic continuation of Mordell-Tornheim and Apostol-Vu $L$-functions, (in Japanese), Master Thesis, Nagoya Univercity, 2003.,
• Xia Zhou, Tianxin Cai and D. Bradley, Signed q-Analogs of Tornheim's Double Series, to appear in Proceedings of the American Mathematical, Society.
• D. Zagier, Values of zeta functions and their applications, First Eupopean Congress of Math., Paris, vol. II, Progress in Math. 120, Birkhäuser 1994, 497–512.