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 i(s), the (Hurwitz's) generalized Zeta function
i(s; a), the Polygamma function / (p) (z) (p = 0; 1; 2; \Delta \Delta \Delta ), and the polylogarithmic function
Li s (z). The key ingredients in our approach include certain known integral representations
for i(s) and i(s; a). The method developed in this paper is illustrated by numerous
examples of closedform 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.
1991 Mathematics Subject Classification. Primary 11M06, 11M35; Secondary 33B15.
Key Words. Zeta functions, Polygamma functions, Polylogarithmic functions, integral
representations, Bernoulli numbers, Stirling numbers, analytic continuation, harmonic
sums, Khintchine constant.
1. Introductions, Definitions, and Preliminaries
A rather classical (over two centuries old) theorem of Christian Goldbach (1690--1764),
which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700--1782),
was revived in 1986 by Shallit and Zikan  as the following problem: