Double Lerch Value Relations and Functional Relations for Witten Zeta Functions

Tokyo Journal of Mathematics

Volumn 31, Issue 2, 2008, Pages 551-574

  Nakamura, Takashi ^a  

^a KYUSHU UNIVERSITY   (Japan)

Author keywords

Double zeta; Functional relation; L and Lerch values; Witten zeta function

Indexed keywords

EID: 77249115032     PISSN: 03873870     EISSN: None     Source Type: Journal    
DOI: 10.3836/tjm/1233844070     Document Type: Article

References (30)

1. T. M. APOSTOL, Introduction to Analytic Number Theory, Springer, 1976.
2. T. ARAKAWA and M. KANEKO, On multiple L-values, J. Math. Soc. Japan 56 (2004), no. 4, 967–991.
3. 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.
4. J. G. HUARD, K. S. WILLIAMS and Z. Y. ZHANG, On Tornheim’s double series, Acta Arith. 75 (1996), 239–252.
5. 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.
6. Y. KOMORI, K. MATSUMOTO and H. TSUMURA, OnWitten multiple zeta-functions associated with semisimple Lie algebras III, preprint.
7. 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.
8. 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.
9. 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.
10. K. MATSUMOTO and H. TSUMURA, A new method of producing functional relations among multiple zeta functions, to appear in Quart. J. Math.
11. 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.
12. L. J. MORDELL, On the evaluation of some multiple series, J. London Math. Soc. 33 (1958), 368–371.
13. T. NAKAMURA, A functional relations for the Tornheim double zeta function, Acta Arith 125 (2006), 257–263.
14. T. NAKAMURA, Double Lerch series and their functional relations, to appear in Aequationes Mathematicae.
15. T. NAKAMURA, Functional relations related to Witten zeta functions, preprint.
16. D. TERHUNE, Evaluation of double L-values, J. Number Theory 105 (2004), no. 2, 275–301.
17. L. TORNHEIM, Harmonic double series, Amer. J. Math. 72 (1950), 303–314.
18. H. TSUMURA, On alternating analogues of Tornheim’s double series, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3633–3641.
19. H. TSUMURA, Evaluation formulae for Tornheim’s type of alternating double series, Math. Comp. 73 (2004), no. 245, 251–258.
20. H. TSUMURA, On Witten’s type of zeta values attached to SO(5), Arch. Math. (Basel) 82 (2004), no. 2, 147–152.
21. H. TSUMURA, On evaluation formulae for double L-values, Bull. Austral. Math. Soc. 70 (2004), no. 2, 213–221.
22. 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.
23. H. TSUMURA, On some functional relations between Mordell-Tornheim double L-functions and Dirichlet L-functions, J. Number Theory 120 (2006), 161–178.
24. H. TSUMURA, On alternating analogues of Tornheim’s double series II, preprint.
25. 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.
26. H. TSUMURA, Certain functional relations for double L-functions, preprint.
27. E. WITTEN, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), no. 1, 153–209.
28. MAOXIANG WU, On analytic continuation of Mordell-Tornheim and Apostol-Vu L-functions, (in Japanese), Master Thesis, Nagoya Univercity, 2003.
29. XIA ZHOU, Tianxin Cai and D. Bradley, Signed q-Analogs of Tornheim’s Double Series, to appear in Proceedings of the American Mathematical Society.
30. 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.