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ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS V

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

Published online by Cambridge University Press:  26 August 2014

YASUSHI KOMORI ,
KOHJI MATSUMOTO  and
HIROFUMI TSUMURA
YASUSHI KOMORI
Affiliation:
Department of Mathematics, Rikkyo University, Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan e-mails: komori@rikkyo.ac.jp
KOHJI MATSUMOTO
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan e-mails: kohjimat@math.nagoya-u.ac.jp
HIROFUMI TSUMURA
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan e-mails: tsumura@tmu.ac.jp
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Abstract

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We study the values of the zeta-function of the root system of type G2 at positive integer points. In our previous work we considered the case when all integers are even, but in the present paper we prove several theorems which include the situation when some of the integers are odd. The underlying reason why we may treat such cases, including odd integers, is also discussed.

MSC classification

Primary: 11M41: Other Dirichlet series and zeta functions
Secondary: 17B20: Simple, semisimple, reductive (super)algebras 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 1 , January 2015 , pp. 107 - 130
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Apostol, T. M., Introduction to analytic number theory (Springer, New York, NY, 1976).Google Scholar
2.Bourbaki, N., Groupes et algèbres de Lie, chapitres 4, 5 et 6 (Hermann, Paris, France, 1968).Google Scholar
3.Humphreys, J. E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics 9 (Springer-Verlag, New York, NY, 1972).CrossRefGoogle Scholar
4.Humphreys, J. E., Reflection groups and coxeter groups (Cambridge University Press, Cambridge, UK, 1990).CrossRefGoogle Scholar
5.Komori, Y., Matsumoto, K. and Tsumura, H., Zeta-functions of root systems, in Proceedings of the conference on L-functions, Fukuoka, Japan, 2006) (Weng, L. and Kaneko, M., Editors) (World Science Publisher, Hackensack, NJ, 2007), 115–140.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., On Witten multiple zeta-functions associated with semisimple Lie algebras II, J. Math. Soc. Japan 62 (2010), 355–394.Google Scholar
8.Komori, Y., Matsumoto, K. and Tsumura, H., On multiple Bernoulli polynomials and multiple L-functions of root systems, Proc. London Math. Soc. 100 (2010), 303–347.Google Scholar
9.Komori, Y., Matsumoto, K. and Tsumura, H., Functional relations for zeta-functions of root systems, in Number theory: Dreaming in dreams – proceedings of the 5th China-Japan seminar (Aoki, T., Kanemitsu, S. and Liu, J.-Y., Editors) (World Science Publisher, Hackensack, NJ, 2010), 135–183.Google Scholar
10.Komori, Y., Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras III, in Multiple Dirichlet series, L-functions and automorphic forms (Bump, D., Friedberg, S. and Goldfeld, D., Editors), Progress in Mathematics Series, vol. 300 (Birkhäuser/Springer, New York, NY, 2012) 223–286.Google Scholar
11.Komori, Y., Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras IV, Glasgow Math. J. 53 (2011), 185–206.CrossRefGoogle Scholar
12.Komori, Y., Matsumoto, K. and Tsumura, H., Zeta-functions of weight lattices of compact connected semisimple Lie groups, preprint, arXiv:math/1011.0323.Google Scholar
13.Matsumoto, K., Nakamura, T., Ochiai, H. and Tsumura, H., On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions, Acta Arith. 132 (2008), 99–125.Google Scholar
14.Matsumoto, K., Nakamura, T. and Tsumura, H., Functional relations and special values of Mordell-Tornheim triple zeta and L-functions, Proc. Amer. Math. Soc. 136 (2008), 2135–2145.CrossRefGoogle Scholar
15.Matsumoto, K. and Tsumura, H., On Witten multiple zeta-functions associated with semisimple Lie algebras I, Ann. Inst. Fourier 56 (2006), 1457–1504.Google Scholar
16.Nakamura, T., A functional relation for the Tornheim double zeta function, Acta Arith. 125 (2006), 257–263.Google Scholar
17.Nakamura, T., Double Lerch series and their functional relations, Aequationes Math. 75 (2008), 251–259.CrossRefGoogle Scholar
18.Nakamura, T., Double Lerch value relations and functional relations for Witten zeta functions, Tokyo J. Math. 31 (2008), 551–574.CrossRefGoogle Scholar
19.Okamoto, T., Multiple zeta values related with the zeta-function of the root system of type A 2, B 2 and G 2, Comment. Math. Univ. St. Pauli 61 (2012), 9–27.Google Scholar
20.Onodera, K., Generalized log sine integrals and the Mordell–Tornheim zeta values, Trans. Amer. Math. Soc. 363 (2011), 1463–1485.Google Scholar
21.Tornheim, L., Harmonic double series, Amer. J. Math. 72 (1950), 303–314.CrossRefGoogle Scholar
22.Tsumura, H., On Witten's type of zeta values attached to SO(5), Arch. Math. (Basel) 82 (2004), 147–152.CrossRefGoogle Scholar
23.Witten, E., On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), 153–209.Google Scholar
24.Zagier, D., Values of zeta functions and their applications, in First European Congress of Mathematics vol. II (Joseph, A.et al. Editors), Progress in Mathematics Series, vol. 120 (Birkhäuser, Basel, Switzerland, 1994), 497–512.Google Scholar
25.Zagier, D., Introduction to multiple zeta values, Lectures at Kyushu University (1999, unpublished note).Google Scholar
26.Zhao, J., Multi-polylogs at twelfth roots of unity and special values of Witten multiple zeta function attached to the exceptional Lie algebra $\mathfrak{g}_2$, J. Algebra Appl. 9 (2010), 327–337.Google Scholar