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

Kohji Matsumoto[1]; Hirofumi Tsumura[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)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 5, page 1457-1504
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Matsumoto, Kohji, and Tsumura, Hirofumi. "On Witten multiple zeta-functions associated with semisimple Lie algebras I." Annales de l’institut Fourier 56.5 (2006): 1457-1504. <http://eudml.org/doc/10182>.

@article{Matsumoto2006,
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.},
affiliation = {Nagoya University Graduate School of Mathematics Chikusa-ku, Nagoya 464-8602 (Japan); Tokyo Metropolitan University Department of Mathematics 1-1, Minami-Ohsawa Hachioji-shi, Tokyo 192-0397 (Japan)},
author = {Matsumoto, Kohji, Tsumura, Hirofumi},
journal = {Annales de l’institut Fourier},
keywords = {Witten multiple zeta-functions; Mordell-Tornheim zeta-functions; Riemann zeta-function; analytic continuation; semisimple Lie algebra},
language = {eng},
number = {5},
pages = {1457-1504},
publisher = {Association des Annales de l’institut Fourier},
title = {On Witten multiple zeta-functions associated with semisimple Lie algebras I},
url = {http://eudml.org/doc/10182},
volume = {56},
year = {2006},
}

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
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 5
SP - 1457
EP - 1504
AB - 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.
LA - eng
KW - Witten multiple zeta-functions; Mordell-Tornheim zeta-functions; Riemann zeta-function; analytic continuation; semisimple Lie algebra
UR - http://eudml.org/doc/10182
ER -

References

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  1. S. Akiyama, S. Egami, Y. Tanigawa, Analytic continuation of multiple zeta-functions and their values at non-positive integers, Acta Arith. 98 (2001), 107-116 Zbl0972.11085MR1831604
  2. M. Bigotte, G. Jacob, N. E. Oussous, M. Petitot, Tables des relations de la fonction zeta colorée avec 1 racine, (1998) 
  3. M. Bigotte, G. Jacob, N. E. Oussous, M. Petitot, Lyndon words and shuffle algebras for generating the coloured multiple zeta values relations tables, WORDS (Rouen, 1999), Theoret. Comput. Sci. 273 (2002), 271-282 Zbl1014.68126MR1872454
  4. J. M. Borwein, R. Girgensohn, Evaluation of triple Euler sums, Elect. J. Combi. 3 (1996) Zbl0884.40005MR1401442
  5. D. Essouabri, Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications à la théorie analytique des nombres, (1995) Zbl0882.11051
  6. D. Essouabri, Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres, Ann. Inst. Fourier 47 (1997), 429-483 Zbl0882.11051MR1450422
  7. P. E. Gunnells, R. Sczech, Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L -functions, Duke Math. J. 118 (2003), 229-260 Zbl1047.11041MR1980994
  8. J. G. Huard, K. S. Williams, N.-Y. Zhang, On Tornheim’s double series, Acta Arith. 75 (1996), 105-117 Zbl0858.40008MR1379394
  9. A. Ivić, The Riemann zeta-function, (1985), Wiley Zbl0556.10026MR792089
  10. A. W. Knapp, Representation theory of semisimple groups, (1986), Princeton University Press, Princeton and Oxford Zbl0604.22001MR855239
  11. K. Matsumoto, On the analytic continuation of various multiple zeta-functions, Number Theory for the Millennium II, Proc. Millennial Conference on Number Theory (2002), 417-440, A K Peters Zbl1031.11051MR1956262
  12. K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I, J. Number Theory 101 (2003), 223-243 Zbl1083.11057MR1989886
  13. K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math J. 172 (2003), 59-102 Zbl1060.11053MR2019520
  14. K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, Proceedings of the Session in analytic number theory and Diophantine equations (Bonn, January-June 2002) 360 (2003), Heath-BrownD. R.D. R. Zbl1056.11049MR2072675
  15. K. Matsumoto, 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), 153-173, Springer Zbl1197.11120MR2213834
  16. K. Matsumoto, H. Tsumura, Generalized multiple Dirichlet series and generalized multiple polylogarithms Zbl1159.11029
  17. L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc. 33 (1958), 368-371 Zbl0081.27501MR100181
  18. H. Samelson, Notes on Lie algebras, (1990), Springer Zbl0708.17005MR1056083
  19. M. V. Subbarao, R. Sitaramachandrarao, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119 (1985), 245-255 Zbl0573.10026MR797027
  20. L. Tornheim, Harmonic double series, Amer. J. Math. 72 (1950), 303-314 Zbl0036.17203MR34860
  21. H. Tsumura, On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function Zbl1136.11056
  22. H. Tsumura, Evaluation formulas for Tornheim’s type of alternating double series, Math. Comp. 73 (2004), 251-258 Zbl1094.11034MR2034120
  23. H. Tsumura, On Witten’s type of zeta values attached to S O ( 5 ) , Arch. Math. (Basel) 84 (2004), 147-152 Zbl1063.40004MR2047668
  24. H. Tsumura, Certain functional relations for the double harmonic series related to the double Euler numbers, J. Austral. Math. Soc., Ser. A 79 (2005), 319-333 Zbl1097.11048MR2190684
  25. E. Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), 153-209 Zbl0762.53063MR1133264
  26. D. Zagier, Values of zeta functions and their applications, Proc. First Congress of Math., Paris, vol.II 120 (1994), 497-512, Birkhäuser Zbl0822.11001MR1341859

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