Summary: SOME SERIES OF THE ZETA AND RELATED FUNCTIONS
Victor S. Adamchik and H.M. Srivastava
Abstract. We propose and develop yet another approach to the problem of summation of
series involving the Riemann Zeta function
s, the Hurwitz's generalized Zeta function
s; a, the Polygamma function p
z p = 0; 1; 2; , and the polylogarithmic function
Lisz. The key ingredients in our approach include certain known integral representations
s; a. The method developed in this paper is illustrated by numerous
examples of closed-form evaluations of series of the aforementioned types; the method
developed in Section 2, in particular, has been implemented in Mathematica Version 3.0.
Many of the resulting summation formulas are believed to be new.
1991Mathematics Subject Classi cation. Primary11M06, 11M35; Secondary 33B15.
Key Words. Zeta functions, Polygamma functions, Polylogarithmic functions, integral
representations, Bernoulli numbers,Stirling numbers, analytic continuation, harmonic
sums, Khintchine constant.