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Some series of the zeta and related functions. (English) Zbl 0919.11056
We propose and develop yet another approach to the problem of summation of series involving the Riemann zeta-function $$\zeta(s)$$, the (Hurwitz’s) generalized zeta-function $$\zeta(s,a)$$, the Polygamma function $$\psi^{(p)} (z)(p=0,1,2,\dots)$$ and the polylogarithmic function $$Li_s(z)$$. The key ingredients in our approach include certain known integral representations for $$\zeta(s)$$ and $$\zeta(s,a)$$. The method 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.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M35 Hurwitz and Lerch zeta functions 33B15 Gamma, beta and polygamma functions
Mathematica
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