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ALTERNATING EULER SUMS AND SPECIAL VALUES OF THE WITTEN MULTIPLE ZETA FUNCTION ATTACHED TO

Part of: Zeta and $L$-functions: analytic theory Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  18 February 2011

JIANQIANG ZHAO
JIANQIANG ZHAO*
Affiliation:
Department of Mathematics, Eckerd College, St Petersburg FL 33711, USA (email: zhaoj@eckerd.edu)
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Abstract

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We study the Witten multiple zeta function associated with the Lie algebra . Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w at least 2 is a finite ℚ-linear combination of alternating Euler sums of weight w and depth at most 2, except when the only nonzero argument is one of the two last variables, in which case ζ(w−1) is needed.

Keywords

Witten zeta functionsWitten volume formulasalternating Euler sums

MSC classification

Secondary: 11M41: Other Dirichlet series and zeta functions 17B20: Simple, semisimple, reductive (super)algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 3 , December 2010 , pp. 419 - 430
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Borwein, J., Lisonek, P. and Irvine, P., ‘An interface for evaluation of Euler sums’, available online at http://oldweb.cecm.sfu.ca/cgi-bin/EZFace/zetaform.cgi.Google Scholar
[2]Bradley, D. M. and Zhou, X., ‘On Mordell–Tornheim sums and multiple zeta values’, Ann. Sci. Math. Québec 34 (2010), 15–23.Google Scholar
[3]Essouabri, D., ‘Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres’, Ann. Inst. Fourier (Grenoble) 47 (1997), 429–483.CrossRefGoogle Scholar
[4]Ihara, K., Kaneko, M. and Zagier, D., ‘Derivation and double shuffle relations for multiple zeta values’, Compositio Math. 142 (2006), 307–338.CrossRefGoogle Scholar
[5]Komori, Y., Matsumoto, K. and Tsumura, H., ‘On Witten multiple zeta-functions associated with semisimple Lie algebras III’, Proc. Conf. on Multiple Dirichlet Series and Applications to Automorphic Forms, Edinburgh, 2008, to appear. Preprint, arxiv: 0907.0955.Google Scholar
[6]Komori, Y., Matsumoto, K. and Tsumura, H., ‘Zeta and L-functions and Bernoulli polynomials of root systems’, Proc. Japan Acad. Ser. A 84 (2008), 57–62.CrossRefGoogle Scholar
[7]Komori, Y., Matsumoto, K. and Tsumura, H., ‘Functional equations for zeta-functions of root systems’, in: Number Theory: Dreaming in Dreams, (Osaka, 2008), (eds. Aoki, T.et al.) (World Scientific, Singapore, 2009), pp. 135–183.CrossRefGoogle Scholar
[8]Komori, Y., Matsumoto, K. and Tsumura, H., ‘On Witten multiple zeta-functions associated with semisimple Lie algebras II’, J. Math. Soc. Japan 62 (2010), 355–394.CrossRefGoogle Scholar
[9]Matsumoto, K., ‘On Mordell–Tornheim and other multiple zeta-functions’, in: Proc. of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, 360 (2003), article number 25, 17 pages.Google Scholar
[10]Mordell, L. J., ‘On the evaluation of some multiple series’, J. Lond. Math. Soc. 33 (1958), 368–371.CrossRefGoogle Scholar
[11]Nielsen, N., Die Gammafunktion. Band I. Handbuch der Theorie der Gammafunktion. Band II. Theorie des Integrallogarithmus und verwandter Transzendenten (Chelsea, New York, 1965).Google Scholar
[12]Terasoma, T., ‘Rational convex cones and cyclotomic multiple zeta values’, Preprint, arXiv: math/0410306.Google Scholar
[13]Tornheim, L., ‘Harmonic double series’, Amer. J. Math. 72 (1950), 303–314.CrossRefGoogle Scholar
[14]Tsumura, H., ‘Combinatorial relations for Euler–Zagier sums’, Acta Arith. 111 (2004), 27–42.CrossRefGoogle Scholar
[15]Tsumura, H., ‘On Witten’s type of zeta values attached to SO(5)’, Arch. Math. (Basel) 82 (2004), 147–152.CrossRefGoogle Scholar
[16]Witten, E., ‘On quantum gauge theories in two-dimensions’, Comm. Math. Phys. 141 (1991), 153–209.CrossRefGoogle Scholar
[17]Zagier, D., Values of zeta function and their applications, in: Proc. of the First European Congress of Math. Vol. 2. (Birkhäuser, Basel–Boston–Berlin, 1994), pp. 497–512.CrossRefGoogle Scholar
[18]Zhao, J., ‘Multiple polylogarithm values at roots of unity’, C. R. Acad. Sci. Paris Ser. I 346 (2008), 1029–1032.CrossRefGoogle Scholar
[19]Zhao, J., ‘Integral structures of multi-polylogs at roots of unity’, Preprint, 2010.Google Scholar
[20]Zhao, J., ‘Witten volume formulas for semi-simple Lie algebras’, Integers, arXiv:1001.3630, in press.Google Scholar
[21]Zhao, J., ‘On a conjecture of Borwein, Bradley and Broadhurst’, J. Reine Angew. Math. 639 (2010), 223–233.Google Scholar
[22]Zhao, J., ‘Multi-polylogs at twelfth roots of unity and special values of Witten multiple zeta function attached to the exceptional Lie algebra 𝔤2’, J. Algebra Appl. 9 (2010), 327–337.CrossRefGoogle Scholar
[23]Zhao, J., ‘Standard relations of multiple polylogarithm values at roots of unity’, Doc. Math. 15 (2010), 1–34.CrossRefGoogle Scholar
[24]Zhao, J. and Zhou, X., ‘Witten multiple zeta values attached to ’, Tokyo J. Math., Preprint MPIM2009-41, 2009, arXiv:0903.2383, in press.Google Scholar