On the values of a class of Dirichlet series at rational arguments

Chakraborty, K.; Kanemitsu, S.; Li, H.-L.

doi:10.1090/S0002-9939-09-10171-5

On the values of a class of Dirichlet series at rational arguments
HTML articles powered by AMS MathViewer

by K. Chakraborty, S. Kanemitsu and H.-L. Li PDF

Proc. Amer. Math. Soc. 138 (2010), 1223-1230 Request permission

Abstract:

In this paper we shall prove that the combination of the general distribution property and the functional equation for the Lipschitz-Lerch transcendent capture the whole spectrum of deeper results on the relations between the values at rational arguments of functions of a class of zeta-functions. By Theorem 1 and its corollaries, we can cover all the previous results in a rather simple and lucid way. By considering the limiting cases, we can also deduce new striking identities for Milnor’s gamma functions, among which is the Gauss second formula for the digamma function.

References

Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
Djurdje Cvijović and Jacek Klinowski, New formulae for the Bernoulli and Euler polynomials at rational arguments, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1527–1535. MR 1283544, DOI 10.1090/S0002-9939-1995-1283544-0
Djurdje Cvijović and Jacek Klinowski, Closed-form summation of some trigonometric series, Math. Comp. 64 (1995), no. 209, 205–210. MR 1270616, DOI 10.1090/S0025-5718-1995-1270616-8
S. Kanemitsu, On evaluation of certain limits in closed form, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 459–474. MR 1024583
Shigeru Kanemitsu and Haruo Tsukada, Vistas of special functions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. MR 2352572, DOI 10.1142/9789812708830
Leonard Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. MR 618278
H.-L. Li, M. Hashimoto and S. Kanemitsu, On structural elucidation of Eisenstein’s formula, to appear.
John Milnor, On polylogarithms, Hurwitz zeta functions, and the Kubert identities, Enseign. Math. (2) 29 (1983), no. 3-4, 281–322. MR 719313
M. Ram Murty and N. Saradha, Transcendental values of the digamma function, J. Number Theory 125 (2007), no. 2, 298–318. MR 2332591, DOI 10.1016/j.jnt.2006.09.017
H. M. Srivastava, A note on the closed-form summation of some trigonometric series, Kobe J. Math. 16 (1999), no. 2, 177–182. MR 1745023
H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 1, 77–84. MR 1757780, DOI 10.1017/S0305004100004412
H. M. Srivastava and Junesang Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, 2001. MR 1849375, DOI 10.1007/978-94-015-9672-5
Yoshihiko Yamamoto, Dirichlet series with periodic coefficients, Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976) Japan Soc. Promotion Sci., Tokyo, 1977, pp. 275–289. MR 0450213