On Witten multiple zeta-functions associated with semisimple Lie algebras I
Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1457-1504.

We define Witten multiple zeta-functions associated with semisimple Lie algebras 𝔰𝔩(n), (n=2,3,...) 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 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.

Nous définissons les fonctions zeta multiples de Witten associées aux algèbres de Lie semi-simples 𝔰𝔩(n), (n=2,3,...), 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 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.

DOI: 10.5802/aif.2218
Classification: 11M41, 40B05
Keywords: Witten multiple zeta-functions, Mordell-Tornheim zeta-functions, Riemann zeta-function, analytic continuation, semisimple Lie algebra
Mot clés : fonctions zeta multiples de Witten, fonctions zeta de Mordell-Tornheim, fonctions zeta de Riemann, suite analytique, algèbre de Lie semisimple

Matsumoto, Kohji 1; Tsumura, Hirofumi 2

1 Nagoya University Graduate School of Mathematics Chikusa-ku, Nagoya 464-8602 (Japan)
2 Tokyo Metropolitan University Department of Mathematics 1-1, Minami-Ohsawa Hachioji-shi, Tokyo 192-0397 (Japan)
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Matsumoto, Kohji; Tsumura, Hirofumi. On Witten multiple zeta-functions associated with semisimple Lie algebras I. Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1457-1504. doi : 10.5802/aif.2218. https://aif.centre-mersenne.org/articles/10.5802/aif.2218/

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

[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., Volume 273 (2002) no. 1-2, pp. 271-282 | DOI | MR | Zbl

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

[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

[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, Volume 47 (1997), pp. 429-483 | DOI | Numdam | MR | Zbl

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

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

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

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

[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 (2002), pp. 417-440 | MR | Zbl

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

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

[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), Volume 360 (2003) no. 25, pp. 17pp | MR | Zbl

[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 (2006), pp. 153-173 | MR

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

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

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

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

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

[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

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

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

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

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

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

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