×

Riemann zeta function: Rapidly converging series and integral representations. (English) Zbl 0746.11034

This paper follows an earlier work [P. L. Butzer, C. Markett and M. Schmidt, Result. Math. 19, 257-274 (1991; Zbl 0724.11036)] on Stirling numbers, central factorial numbers, and their relation to the values of the Riemann zeta function \(\zeta(m)\) for integer \(m\geq 2\), with special emphasis on the interaction between sum and integral identities. The principal results are new rapidly convergent series and integral representations for \(\zeta(m)\) and for special Dirichlet \(L\)-functions with integer argument.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M41 Other Dirichlet series and zeta functions

Citations:

Zbl 0724.11036
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Butzer, P. L.; Markett, C.; Schmidt, M., Stirling numbers, central factorial numbers, and representations of the Riemann Zeta function, Resultate Math., 19, 257-274 (1991) · Zbl 0724.11038
[2] Butzer, P. L.; Schmidt, M.; Stark, E. L.; Vogt, L., Central factorial numbers; their main properties and some applications, Numer. Funct. Anal. Optim., 10, 419-488 (1989) · Zbl 0659.10012
[3] Apéry, R., Irrationalité de ζ(2) et ζ(3), Journées arithmétiques de Luminy, Astérique, 61, 11-13 (1979) · Zbl 0401.10049
[4] Catalan, E., Mémoire sur la transformation des séries et sur quelques intégrales défines, Mémoires Académie Royale de Belgique, 33, 1-50 (1865)
[5] Slater, L. J., Generalized Hypergeometric Functions (1966), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0135.28101
[6] P.L. Butzer and M. Hauss, Integral and rapidly converging series representations of the Dirichlet \(LL_1sL_{-4}s\); P.L. Butzer and M. Hauss, Integral and rapidly converging series representations of the Dirichlet \(LL_1sL_{-4}s\) · Zbl 0760.11022
[7] Lewin, L., Polylogarithms and Associated Functions (1981), North-Holland: North-Holland New York · Zbl 0465.33001
[8] Butzer, P. L.; Hauss, M.; Schmidt, M., Factorial functions and Stirling numbers of fractional orders, Resultate Math., 16, 16-48 (1989) · Zbl 0707.05002
[9] Butzer, P. L.; Hauss, M., Stirling functions of first and second kind, ((Proc. Conf. in honour of Prof. Jakimovski, Research Inst. of Math. Sciences, Bar Ilan University, Tel Aviv, June 4-8, 1990). (Proc. Conf. in honour of Prof. Jakimovski, Research Inst. of Math. Sciences, Bar Ilan University, Tel Aviv, June 4-8, 1990), Approximation, Interpolation and Summability, Vol. 4 (1991), Weizmann Press: Weizmann Press Israel), 89-108, Some new applications · Zbl 0809.39009
[10] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1964), Dover Publications, Inc: Dover Publications, Inc New York · Zbl 0515.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.