We define Witten multiple zeta-functions associated with semisimple Lie algebras , 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 , 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 , , 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 , 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.
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
@article{AIF_2006__56_5_1457_0, author = {Matsumoto, Kohji and Tsumura, Hirofumi}, title = {On {Witten} multiple zeta-functions associated with semisimple {Lie} algebras {I}}, journal = {Annales de l'Institut Fourier}, pages = {1457--1504}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {5}, year = {2006}, doi = {10.5802/aif.2218}, mrnumber = {2273862}, zbl = {1168.11036}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2218/} }
TY - JOUR AU - Matsumoto, Kohji AU - Tsumura, Hirofumi TI - On Witten multiple zeta-functions associated with semisimple Lie algebras I JO - Annales de l'Institut Fourier PY - 2006 SP - 1457 EP - 1504 VL - 56 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2218/ DO - 10.5802/aif.2218 LA - en ID - AIF_2006__56_5_1457_0 ER -
%0 Journal Article %A Matsumoto, Kohji %A Tsumura, Hirofumi %T On Witten multiple zeta-functions associated with semisimple Lie algebras I %J Annales de l'Institut Fourier %D 2006 %P 1457-1504 %V 56 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2218/ %R 10.5802/aif.2218 %G en %F AIF_2006__56_5_1457_0
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] 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] Tables des relations de la fonction zeta colorée avec 1 racine (1998) (Prépublication du LIFL, USTL)
[3] 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] Evaluation of triple Euler sums, Elect. J. Combi., Volume 3 (1996) no. 1, pp. 27 (R23) | MR | Zbl
[5] 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] 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] Evaluation of Dedekind sums, Eisenstein cocycles, and special values of -functions, Duke Math. J., Volume 118 (2003), pp. 229-260 | DOI | MR | Zbl
[8] On Tornheim’s double series, Acta Arith., Volume 75 (1996), pp. 105-117 | MR | Zbl
[9] The Riemann zeta-function, Wiley, 1985 | MR | Zbl
[10] Representation theory of semisimple groups, Princeton University Press, Princeton and Oxford, 1986 | MR | Zbl
[11] 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] 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] 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] 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] 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] Generalized multiple Dirichlet series and generalized multiple polylogarithms (Acta Arith., to appear) | Zbl
[17] On the evaluation of some multiple series, J. London Math. Soc., Volume 33 (1958), pp. 368-371 | DOI | MR | Zbl
[18] Notes on Lie algebras, Universitext, Springer, 1990 | MR | Zbl
[19] On some infinite series of L. J. Mordell and their analogues, Pacific J. Math., Volume 119 (1985), pp. 245-255 | MR | Zbl
[20] Harmonic double series, Amer. J. Math., Volume 72 (1950), pp. 303-314 | DOI | MR | Zbl
[21] On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function (Math. Proc. Cambridge, Philos. Soc., to appear) | Zbl
[22] Evaluation formulas for Tornheim’s type of alternating double series, Math. Comp., Volume 73 (2004), pp. 251-258 | DOI | MR | Zbl
[23] On Witten’s type of zeta values attached to , Arch. Math. (Basel), Volume 84 (2004), pp. 147-152 | MR | Zbl
[24] 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] On quantum gauge theories in two dimensions, Comm. Math. Phys., Volume 141 (1991), pp. 153-209 | DOI | MR | Zbl
[26] 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
Cited by Sources: