Proceedings of the Japan Academy, Series A, Mathematical Sciences

Zeta and $L$-functions and Bernoulli polynomials of root systems

Abstract

This article is essentially an announcement of the papers [7,8,9,10] of the authors, though some of the examples are not included in those papers. We consider what is called zeta and $L$-functions of root systems which can be regarded as a multi-variable version of Witten multiple zeta and $L$-functions. Furthermore, corresponding to these functions, Bernoulli polynomials of root systems are defined. First we state several analytic properties, such as analytic continuation and location of singularities of these functions. Secondly we generalize the Bernoulli polynomials and give some expressions of values of zeta and $L$-functions of root systems in terms of these polynomials. Finally we give some functional relations among them by our previous method. These relations include the known formulas for their special values formulated by Zagier based on Witten’s work.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 84, Number 5 (2008), 57-62.

Dates
First available in Project Euclid: 1 May 2008

https://projecteuclid.org/euclid.pja/1209649653

Digital Object Identifier
doi:10.3792/pjaa.84.57

Mathematical Reviews number (MathSciNet)
MR2415897

Zentralblatt MATH identifier
1147.11053

Citation

Komori, Yasushi; Matsumoto, Kohji; Tsumura, Hirofumi. Zeta and $L$-functions and Bernoulli polynomials of root systems. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 5, 57--62. doi:10.3792/pjaa.84.57. https://projecteuclid.org/euclid.pja/1209649653.

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer, New York, 1976.
• P. E. Gunnells and R. Sczech, Evaluation of Dedekind sums, Eisenstein cocycles, and special values of $L$-functions, Duke Math. J. 118 (2003), 229–260.
• J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972.
• J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Univ. Press, Cambridge, 1990.
• Y. Komori, K. Matsumoto and H. Tsumura, Zeta-functions of root systems, in “Proceedings of the Conference on $L$-functions” (Fukuoka, 2006), L. Weng and M. Kaneko (eds), World Scientific, 2007, pp. 115–140.
• Y. Komori, K. Matsumoto and H. Tsumura, Zeta-functions of root systems, their functional relations, and Dynkin diagrams, in Analytic Number Theory' (Kyoto, 2006), RIMS Kokyuroku. (to appear).
• Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras II. (Preprint).
• Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras III. (Preprint).
• Y. Komori, K. Matsumoto and H. Tsumura, On multiple Bernoulli polynomials and multiple $L$-functions of root systems. (Preprint).
• Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras IV. (in preparation).
• K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math J. 172 (2003), 59–102.
• K. Matsumoto, On the analytic continuation of various multiple zeta-functions, in Number theory for the millennium, II (Urbana, IL, 2000), 417–440, A K Peters, Natick, MA.
• K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I, J. Number Theory 101 (2003), 223–243.
• K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, in Proceedings of the Session in analytic number theory and Diophantine equations' (Bonn, January-June 2002), D. R. Heath-Brown and B. Z. Moroz (eds.), Bonner Mathematische Schriften Nr. 360, Bonn 2003, n.25, 17pp.
• K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras I, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, 1457–1504.
• K. Matsumoto and H. Tsumura, Functional relations for various multiple zeta-functions, in `Analytic Number Theory' (Kyoto, 2005), RIMS Kokyuroku No. 1512 (2006), 179–190.
• T. Nakamura, Double $L$-value relations and functional relations for Witten zeta functions, Tokyo J. Math. (to appear).
• H. Tsumura, On some functional relations between Mordell-Tornheim double L-functions and Dirichlet L-functions, J. Number Theory 120 (2006), no. 1, 161–178.
• H. Tsumura, On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 3, 395–405.
• E. Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), no. 1, 153–209.
• D. Zagier, Values of zeta functions and their applications, in First European Congress of Mathematics, Vol. II (Paris, 1992), 497–512, Progr. Math., 120, Birkhäuser, Basel, 1994.