Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-08-21T16:21:33.421Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A CERTAIN CONVOLUTION OF POLYLOGARITHMS

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  07 February 2012

HIROFUMI TSUMURA
HIROFUMI TSUMURA*
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan (email: tsumura@tmu.ac.jp)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we consider certain double series analogous to Tornheim’s double series and real analytic Eisenstein series. By computing double integrals in two ways, we express the double series as a sum of products of polylogarithms. The technique generalises one given by Kanemitsu, Tanigawa and Yoshimoto. Evaluating the double series at particular points gives new evaluations for certain double series in terms of values of the Riemann zeta function and the dilogarithm which are analogues of formulas of Mordell and Goncharov.

Keywords

polylogarithmsRiemann zeta-functionreal analytic Eisenstein series

MSC classification

Secondary: 11M41: Other Dirichlet series and zeta functions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 461 - 472
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Berndt, B. C., Ramanujan’s Notebooks, Part II (Springer, New York, 1989).CrossRefGoogle Scholar
[2]Cauchy, A., Oeuvres Complètes d’Augustin Cauchy, Série II, t. VII (Gauthier-Villars, Paris, 1889).Google Scholar
[3]Dilcher, K., ‘Zeros of Bernoulli, generalized Bernoulli and Euler polynomials’, Mem. Amer. Math. Soc. 73(386) (1988).Google Scholar
[4]Goncharov, A. B., ‘The double logarithm and Manin’s complex for modular curves’, Math. Res. Lett. 4(5) (1997), 617–636.CrossRefGoogle Scholar
[5]Huard, J. G., Williams, K. S. and Zhang, N.-Y., ‘On Tornheim’s double series’, Acta Arith. 75(2) (1996), 105–117.CrossRefGoogle Scholar
[6]Kanemitsu, S., Tanigawa, Y. and Yoshimoto, M., ‘Convolution of Riemann zeta-values’, J. Math. Soc. Japan 57(4) (2005), 1167–1177.Google Scholar
[7]Komori, Y., Matsumoto, K. and Tsumura, H., ‘On multiple Bernoulli polynomials and multiple L-functions of root systems’, Proc. Lond. Math. Soc. (3) 100(2) (2010), 303–347.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(2) (2010), 355–394.CrossRefGoogle Scholar
[9]Langlands, R. P., On the Functional Equations Satisfied by Eisenstein Series (Springer, Berlin, 1976).Google Scholar
[10]Lewin, L., Polylogarithms and Associated Functions (North-Holland Publishing Co., Amsterdam, 1981).Google Scholar
[11]Matsumoto, K., ‘On the analytic continuation of various multiple zeta-functions’, in: Number Theory for the Millennium, II (Urbana, IL, 2000) (A. K. Peters, Natick, MA, 2002), pp. 417–440.Google Scholar
[12]Matsumoto, K. and Tsumura, H., ‘On Witten multiple zeta-functions associated with semisimple Lie algebras I’, Ann. Inst. Fourier (Grenoble) 56(5) (2006), 1457–1504.CrossRefGoogle Scholar
[13]Matsumoto, K. and Tsumura, H., ‘Functional relations among certain double polylogarithms and their character analogues’, Šiauliai Math. Semin. 3(11) (2008), 189–205.Google Scholar
[14]Mellin, Hj., ‘Eine Formel für den Logarithmus transcendenter Funktionen von endlichem Geschlecht’, Acta Math 25(1) (1902), 165–183.Google Scholar
[15]Mordell, L. J., ‘On the evaluation of some multiple series’, J. Lond. Math. Soc. 33 (1958), 368–371.CrossRefGoogle Scholar
[16]Nakamura, T., ‘A functional relation for the Tornheim double zeta function’, Acta Arith. 125(3) (2006), 257–263.CrossRefGoogle Scholar
[17]Tornheim, L., ‘Harmonic double series’, Amer. J. Math. 72 (1950), 303–314.CrossRefGoogle Scholar
[18]Tsumura, H., ‘On some combinatorial relations for Tornheim’s double series’, Acta Arith. 105(3) (2002), 239–252.Google Scholar
[19]Tsumura, H., ‘On certain polylogarithmic double series’, Arch. Math. (Basel) 88(1) (2007), 42–51.Google Scholar
[20]Tsumura, H., ‘On functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function’, Math. Proc. Cambridge Philos. Soc. 142(3) (2007), 395–405.Google Scholar
[21]Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, 4th edn (Cambridge University Press, Cambridge, 1996).Google Scholar
[22]Zagier, D., ‘Values of zeta functions and their applications’, in: First European Congress of Mathematics,Vol. II (Paris, 1992), Progress in Mathematics, 120 (Birkhäuser, Basel, 1994), pp. 497–512.Google Scholar