previous ::
next
Double Lerch Value Relations and Functional Relations for Witten Zeta Functions
Takashi NAKAMURA
Source: Tokyo J. of Math. Volume 31, Number 2 (2008), 551-574.
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.
Primary Subjects: 11M41
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.tjm/1233844070
Digital Object Identifier: doi:10.3836/tjm/1233844070
Mathematical Reviews number (MathSciNet):
MR2477890
Zentralblatt MATH identifier:
05545430
References
T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.
Mathematical Reviews (MathSciNet):
MR434929
T. Arakawa and M. Kaneko, On multiple $L$-values, J. Math. Soc. Japan, 56 (2004), no. 4, 967--991.
Mathematical Reviews (MathSciNet):
MR2091412
Zentralblatt MATH:
1065.11068
Digital Object Identifier: doi:10.2969/jmsj/1190905444
Project Euclid: euclid.jmsj/1190905444
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.
Mathematical Reviews (MathSciNet):
MR1980994
Zentralblatt MATH:
1047.11041
Digital Object Identifier: doi:10.1215/S0012-7094-03-11822-0
Project Euclid: euclid.dmj/1082744647
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.
Mathematical Reviews (MathSciNet):
MR2310292
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.
Mathematical Reviews (MathSciNet):
MR2075634
Zentralblatt MATH:
1056.11049
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.
Mathematical Reviews (MathSciNet):
MR2395147
Zentralblatt MATH:
1143.11035
Digital Object Identifier: doi:10.4064/aa132-2-1
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..
Mathematical Reviews (MathSciNet):
MR2383519
Zentralblatt MATH:
05288841
Digital Object Identifier: doi:10.1090/S0002-9939-08-09192-2
K. Matsumoto and H. Tsumura, A new method of producing functional relations among multiple zeta functions, to appear in Quart. J., Math.
Mathematical Reviews (MathSciNet):
MR2392501
Zentralblatt MATH:
1151.11045
Digital Object Identifier: doi:10.1093/qmath/ham025
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.
Mathematical Reviews (MathSciNet):
MR2273862
L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc., 33 (1958), 368--371.
Mathematical Reviews (MathSciNet):
MR100181
Digital Object Identifier: doi:10.1112/jlms/s1-33.3.368
T. Nakamura, A functional relations for the Tornheim double zeta function, Acta Arith, 125 (2006), 257--263.
Mathematical Reviews (MathSciNet):
MR2276193
Zentralblatt MATH:
1153.11047
Digital Object Identifier: doi:10.4064/aa125-3-3
T. Nakamura, Double Lerch series and their functional relations, to appear in Aequationes, Mathematicae.
Mathematical Reviews (MathSciNet):
MR2424133
Zentralblatt MATH:
05303964
Digital Object Identifier: doi:10.1007/s00010-007-2921-7
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.
Mathematical Reviews (MathSciNet):
MR2040159
Zentralblatt MATH:
1041.11061
Digital Object Identifier: doi:10.1016/j.jnt.2003.11.007
L. Tornheim, Harmonic double series, Amer. J. Math., 72 (1950), 303--314.
Mathematical Reviews (MathSciNet):
MR34860
Zentralblatt MATH:
0036.17203
Digital Object Identifier: doi:10.2307/2372034
JSTOR: links.jstor.org
H. Tsumura, On alternating analogues of Tornheim's double series, Proc. Amer. Math. Soc., 131 (2003), no. 12, 3633--3641.
Mathematical Reviews (MathSciNet):
MR1998168
Zentralblatt MATH:
1045.11057
Digital Object Identifier: doi:10.1090/S0002-9939-03-07186-7
JSTOR: links.jstor.org
H. Tsumura, Evaluation formulae for Tornheim's type of alternating double series, Math. Comp., 73 (2004), no. 245, 251--258.
Mathematical Reviews (MathSciNet):
MR2034120
Zentralblatt MATH:
1094.11034
Digital Object Identifier: doi:10.1090/S0025-5718-03-01572-2
H. Tsumura, On Witten's type of zeta values attached to $\rm SO(5)$, Arch. Math. (Basel), 82 (2004), no. 2, 147--152.
Mathematical Reviews (MathSciNet):
MR2047668
Zentralblatt MATH:
1063.40004
Digital Object Identifier: doi:10.1007/s00013-003-4809-7
H. Tsumura, On evaluation formulae for double $L$-values, Bull. Austral. Math. Soc., 70 (2004), no. 2, 213--221.
Mathematical Reviews (MathSciNet):
MR2094289
Zentralblatt MATH:
1060.11054
Digital Object Identifier: doi:10.1017/S0004972700034432
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.
Mathematical Reviews (MathSciNet):
MR2190684
Zentralblatt MATH:
1097.11048
Digital Object Identifier: doi:10.1017/S1446788700010922
H. Tsumura, On some functional relations between Mordell-Tornheim double $L$-functions and Dirichlet $L$-functions, J. Number Theory, 120 (2006), 161--178.
Mathematical Reviews (MathSciNet):
MR2256802
Zentralblatt MATH:
1136.11056
Digital Object Identifier: doi:10.1016/j.jnt.2005.11.006
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.
Mathematical Reviews (MathSciNet):
MR2329691
Zentralblatt MATH:
1149.11044
Digital Object Identifier: doi:10.1017/S0305004107000059
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.
Mathematical Reviews (MathSciNet):
MR1133264
Zentralblatt MATH:
0762.53063
Digital Object Identifier: doi:10.1007/BF02100009
Project Euclid: euclid.cmp/1104248198
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.
Mathematical Reviews (MathSciNet):
MR1341859
Zentralblatt MATH:
0822.11001
previous ::
next
2009 © Publication Committee for the Tokyo Journal of Mathematics
Tokyo Journal of Mathematics
