ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS IV

KOMORI, YASUSHI ; MATSUMOTO, KOHJI ; TSUMURA, HIROFUMI

Glasgow mathematical journal, Tome 53 (2011) no. 1, pp. 185-206

Résumé

In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types A2, A3, B2, B3 and C3. In this paper, we consider the case of G2-type. We define certain analogues of Bernoulli polynomials of G2-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of G2-type. Next, we consider the meromorphic continuation of the zeta-function of G2-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.

Détail

DOI : 10.1017/S0017089510000613

Mots-clés : Primary 11M41, Secondary 17B20, 40B05

Bibliographie

[1] 1.Bourbaki, N., Groupes et Algèbres de Lie, Chapitres 4, 5 et 6 (Hermann, Paris, 1968). Google Scholar

[2] 2.Essouabri, D., Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications à la théorie analytique des nombres, Thèse (Université Nancy I, 1995). Google Scholar

[3] 3.Essouabri, D., Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications an théorie analytique des nombres, Ann. Inst. Fourier 47 (1997), 429–483. Google Scholar | DOI

[4] 4.Komori, Y., An integral representation of multiple Hurwitz–Lerch zeta functions and generalized multiple Bernoulli numbers, Qt. J. Math. (Oxford), doi:10.1093/qmath/hap004 (in press). Google Scholar | DOI

[5] 5.Komori, Y., Matsumoto, K. and Tsumura, H., Zeta-functions of root systems, in The Conference on L-functions (Fukuoka, 2006) (Weng, L. and Kaneko, M., Editors) (World Scientific, Hackensack, NJ, 2007), pp. 115–140. Google Scholar

[6] 6.Komori, Y., Matsumoto, K. and Tsumura, H., Zeta and L-functions and Bernoulli polynomials of root systems, Proc. Japan Acad., Ser. A 84 (2008), 57–62. Google Scholar

[7] 7.Komori, Y., Matsumoto, K. and Tsumura, H., Functional relations for zeta-functions of root systems, in Number Theory: Dreaming in Dreams – Proceedings of the 5th China–Japan Seminar (Aoki, T., Kanemitsu, S. and Liu, J.-Y., Editors) (World Scientific, Hackensack, NJ, 2010), 135–183. Google Scholar

[8] 8.Komori, Y., Matsumoto, K. and Tsumura, H., On multiple Bernoulli polynomials and multiple L-functions of root systems, Proc. Lond. Math. Soc. 100 (2010), 303–347. Google Scholar | DOI

[9] 9.Komori, Y., Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras II, J. Math. Soc. Japan 62 (2010), 355–394. Google Scholar | DOI

[10] 10.Komori, Y., Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras III, preprint, arXiv:0907.0955. Google Scholar

[11] 11.Matsumoto, K., On the analytic continuation of various multiple zeta-functions, in Number Theory for the Millennium II, Proc. Millennial Conference on Number Theory (Bennett, M. A. et al. , Editors) (A K Peters, Natick, MA, 2002), 417–440. Google Scholar

[12] 12.Matsumoto, K., Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math J. 172 (2003), 59–102. Google Scholar | DOI

[13] 13.Matsumoto, K., The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I, J. Number Theory 101 (2003), 223–243. Google Scholar | DOI

[14] 14.Matsumoto, K., On Mordell-Tornheim and other multiple zeta-functions, in Proceedings of the Session in Analytic Number Theory and Diophantine Equations (Heath-Brown, D. R. and Moroz, B. Z., Editors), Bonner Mathematische Schriften Nr. 360, (Bonn, Germany, 2003), 17 pp. Google Scholar

[15] 15.Matsumoto, K., Nakamura, T., Ochiai, H. and Tsumura, H., On value-relations, functional relations and singularities of Mordell–Tornheim and related triple zeta-functions, Acta Arith. 132 (2008), 99–125. Google Scholar | DOI

[16] 16.Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras I, Ann. Inst. Fourier 56 (2006), 1457–1504. Google Scholar | DOI

[17] 17.Tsumura, H., On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function, Math. Proc. Camb. Phil. Soc. 142 (2007), 395–405. Google Scholar | DOI

[18] 18.Witten, E., On quantum gauge theories in two dimensions, Commun. Math. Phys. 141 (1991), 153–209. Google Scholar | DOI

[19] 19.Zagier, D., Values of zeta functions and their applications, in First European Congress of Mathematics vol. II (Joseph, A. et al. , Editors), Progress in Mathematics, vol. 120 (Birkhäuser, Basel, 1994), 497–512. Google Scholar

[20] 20.Zhao, J., Multi-polylogs at twelfth roots of unity and special values of Witten multiple zeta function attached to the exceptional Lie algebra , J. Algebra Appl. 9 (2010), 327–337. Google Scholar | DOI