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

Matsumoto, Kohji; Tsumura, Hirofumi

Annales de l'Institut Fourier, Tome 56 (2006) no. 5, pp. 1457-1504.

Nous définissons les fonctions zeta multiples de Witten associées aux algèbres de Lie semi-simples $\mathrm{𝔰𝔩}\left(n\right)$, $(n=2,3,...)$, 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 $\mathrm{𝔰𝔩}\left(4\right)$, 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.

We define Witten multiple zeta-functions associated with semisimple Lie algebras $\mathrm{𝔰𝔩}\left(n\right)$, $(n=2,3,...)$ 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 $\mathrm{𝔰𝔩}\left(4\right)$, 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.

[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, A K Peters (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, Springer (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 $SO\left(5\right)$, 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, Birkhäuser (1994), pp. 497-512 | MR | Zbl