The series is convergent when s is a complex number with <(s) > 1. Some special values of ζ(s) are well known, for example the values ζ(2) = π/6, ζ(4) = π/90, were obtained by Euler. In 1859, Riemann had the idea to define ζ(s) for all complex number s by analytic continuation. This continuation is very important in number theory and plays a central role in the study of the distribution of prim...

In this paper, we obtain several expansions for ζ(s) involving a sequence of polynomials in s, denoted by αk(s). These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some series expansions for the zeta function that are known for integer values of s. The expansions also give a different approach to the analytic continuation of the...