On Witten multiple zeta-functions associated with semisimple Lie algebras I

[Sur les fonctions zeta multiples de Witten associées aux algèbres de Lie semi-simples]

Matsumoto, Kohji ; Tsumura, Hirofumi

Annales de l'Institut Fourier, Tome 56 (2006), p. 1457-1504 / Harvested from Numdam

Résumé

Nous définissons les fonctions zeta multiples de Witten associées aux algèbres de Lie semi-simples ${\mathfrak{sl}}(n)$, $(n=2,3,\ldots )$, et démontrons leurs continuations analytiques. Elles peuvent être considérées comme des généralisations à plusieurs variables des fonctions zeta de Witten définies par Zagier. Dans le cas ${\mathfrak{sl}}(4)$, nous déterminons les singularités de la fonction zeta multiple. De plus, nous démontrons plusieurs relations fonctionnelles entre cette fonction, les fonctions zeta doubles de Mordell-Tornheim et la fonction zeta de Riemann. En utilisant ces relations, nous démontrons de nouvelles formules non-triviales pour évaluer des valeurs spécifiques de cette fonction aux points entiers positifs.

Abstract

We define Witten multiple zeta-functions associated with semisimple Lie algebras ${\mathfrak{sl}}(n)$, $(n=2,3,\ldots )$ of several complex variables, and prove the analytic continuation of them. These can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case ${\mathfrak{sl}}(4)$, we determine the singularities of this function. Furthermore we prove certain functional relations among this function, the Mordell-Tornheim double zeta-functions and the Riemann zeta-function. Using these relations, we prove new and non-trivial evaluation formulas for special values of this function at positive integers.

Détail

Publié le : 2006-01-01

MR 2273862 | Zbl 1168.11036

DOI : https://doi.org/10.5802/aif.2218
Classification:  11M41,  40B05
Mots clés: fonctions zeta multiples de Witten, fonctions zeta de Mordell-Tornheim, fonctions zeta de Riemann, suite analytique, algèbre de Lie semisimple

Bibliographie

[1] Akiyama, S.; Egami, S.; Tanigawa, Y. Analytic continuation of multiple zeta-functions and their values at non-positive integers, Acta Arith., Tome 98 (2001), pp. 107-116 | Article | MR 1831604 | Zbl 0972.11085

[2] Bigotte, M.; Jacob, G.; Oussous, N. E.; Petitot, M. Tables des relations de la fonction zeta colorée avec 1 racine (1998) (Prépublication du LIFL, USTL)

[3] Bigotte, M.; Jacob, G.; Oussous, N. E.; Petitot, M. Lyndon words and shuffle algebras for generating the coloured multiple zeta values relations tables, WORDS (Rouen, 1999), Theoret. Comput. Sci., Tome 273 (2002) no. 1-2, pp. 271-282 | Article | MR 1872454 | Zbl 1014.68126

[4] Borwein, J. M.; Girgensohn, R. Evaluation of triple Euler sums, Elect. J. Combi., Tome 3 (1996) no. 1, pp. 27 (R23) | MR 1401442 | Zbl 0884.40005

[5] Essouabri, D. Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications à la théorie analytique des nombres, Thèse, Univ. Henri Poincaré - Nancy I (1995) (Ph. D. Thesis) | Numdam | Zbl 0882.11051

[6] Essouabri, D. Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres, Ann. Inst. Fourier, Tome 47 (1997), pp. 429-483 | Article | Numdam | MR 1450422 | Zbl 0882.11051

[7] Gunnells, P. E.; Sczech, R. Evaluation of Dedekind sums, Eisenstein cocycles, and special values of $L$-functions, Duke Math. J., Tome 118 (2003), pp. 229-260 | Article | MR 1980994 | Zbl 1047.11041

[8] Huard, J. G.; Williams, K. S.; Zhang, N.-Y. On Tornheim’s double series, Acta Arith., Tome 75 (1996), pp. 105-117 | MR 1379394 | Zbl 0858.40008

[9] Ivić, A. The Riemann zeta-function, Wiley (1985) | MR 792089 | Zbl 0556.10026

[10] Knapp, A. W. Representation theory of semisimple groups, Princeton University Press, Princeton and Oxford (1986) | MR 855239 | Zbl 0604.22001

[11] Matsumoto, K.; Bannett, M. A. On the analytic continuation of various multiple zeta-functions, Number Theory for the Millennium II, Proc. Millennial Conference on Number Theory, A K Peters (2002), pp. 417-440 | MR 1956262 | Zbl 1031.11051

[12] Matsumoto, K. The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I, J. Number Theory, Tome 101 (2003), pp. 223-243 | Article | MR 1989886 | Zbl 01961834

[13] Matsumoto, K. Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math J., Tome 172 (2003), pp. 59-102 | MR 2019520 | Zbl 1060.11053

[14] Matsumoto, K.; Heath-Brown, D. R.; Moroz, B. Z. On Mordell-Tornheim and other multiple zeta-functions, Proceedings of the Session in analytic number theory and Diophantine equations (Bonn, January-June 2002) (Bonner Mathematische Schriften) Tome 360 (2003) no. 25, pp. 17pp | MR 2072675 | Zbl 1056.11049

[15] Matsumoto, K.; Zhang, Wenpeng; Tanigawa, Y. Analytic properties of multiple zeta-functions in several variables, Proceedings of the 3rd China-Japan Seminar (Xi’an 2004) : The Tradition and Modernization in Number Theory, Springer (2006), pp. 153-173 | MR 2213834 | Zbl pre05346678

[16] Matsumoto, K.; Tsumura, H. Generalized multiple Dirichlet series and generalized multiple polylogarithms (Acta Arith., to appear) | Zbl 05082206

[17] Mordell, L. J. On the evaluation of some multiple series, J. London Math. Soc., Tome 33 (1958), pp. 368-371 | Article | MR 100181 | Zbl 0081.27501

[18] Samelson, H. Notes on Lie algebras, Springer, Universitext (1990) | MR 1056083 | Zbl 0708.17005

[19] Subbarao, M. V.; Sitaramachandrarao, R. On some infinite series of L. J. Mordell and their analogues, Pacific J. Math., Tome 119 (1985), pp. 245-255 | MR 797027 | Zbl 0573.10026

[20] Tornheim, L. Harmonic double series, Amer. J. Math., Tome 72 (1950), pp. 303-314 | Article | MR 34860 | Zbl 0036.17203

[21] Tsumura, H. On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function (Math. Proc. Cambridge, Philos. Soc., to appear) | Zbl 05058330

[22] Tsumura, H. Evaluation formulas for Tornheim’s type of alternating double series, Math. Comp., Tome 73 (2004), pp. 251-258 | Article | MR 2034120 | Zbl 01998518

[23] Tsumura, H. On Witten’s type of zeta values attached to $SO(5)$, Arch. Math. (Basel), Tome 84 (2004), pp. 147-152 | MR 2047668 | Zbl 1063.40004

[24] Tsumura, H. Certain functional relations for the double harmonic series related to the double Euler numbers, J. Austral. Math. Soc., Ser. A, Tome 79 (2005), pp. 319-333 | Article | MR 2190684 | Zbl 1097.11048

[25] Witten, E. On quantum gauge theories in two dimensions, Comm. Math. Phys., Tome 141 (1991), pp. 153-209 | Article | MR 1133264 | Zbl 0762.53063

[26] Zagier, D. Values of zeta functions and their applications, Proc. First Congress of Math., Paris, vol.II, Birkhäuser (Progress in Math.) Tome 120 (1994), pp. 497-512 | MR 1341859 | Zbl 0822.11001