The Hurwitz Zeta Function at the Positive Integers

Abstract

A formula for the Hurwitz zeta function at the positive integers $k$, $ζ(k,b)$, is created by solving the real and the imaginary parts separately and then combining them. A few different formulae for the Hurwitz zeta function are known from the literature, but they are very general and usually hold for $\Re{(k)}>1$. The advantage of formulae that only hold at the positive integers is the fact that they are simpler and easier to work with. An analytic continuation of the generating function of $ζ(k,b)$ is also obtained as $\sum_{k\ge 2}x^k(ζ(k,b)-1/b^k)$, where the term $1/b^k$ was subtracted for convenience.