# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Riemann zeta function: Rapidly converging series and integral representations. (English) Zbl 0746.11034
This paper follows an earlier work [{\it P. L. Butzer, C. Markett} and {\it 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 $m\ge 2$, 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
Full Text:
##### 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 ${\zeta}(2)$ et ${\zeta}$(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) · Zbl 0135.28101 [6] P.L. Butzer and M. Hauss, Integral and rapidly converging series representations of the Dirichlet L-functions L1(s) and L-4(s), Atti Sem. Mat. Fis. Univ. Modena (to appear). · Zbl 0760.11022 [7] Lewin, L.: Polylogarithms and associated functions. (1981) · 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. Approximation, interpolation and summability 4, 89-108 (1991) · Zbl 0809.39009 [10] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions. (1964) · Zbl 0171.38503