A note on the Hurwitz zeta function

Abstract

We show that the Hurwitz zeta function and polylogarithm, $\zeta(\nu,a)$ and $\operatorname{Li}_{\nu}(z)$, form a discrete Fourier transform pair for $\Re\nu>1$. Many formulae, the majority of them previously unknown, are obtained as a corollary to this result. In particular, the transformation relation allows the evaluation of $\zeta(\nu,a)$ at rational values of the parameter~$a$. It is also shown that, by making use of the transform pair, various known results can be deduced easily and in a unified manner. For instance, $$2\zeta(2n+1,1/3)=(3^{2n+1}-1)\zeta(2n+1)+(-1)^{n-1}3^{2n}\sqrt3\dfrac{(2\pi)^{2n+1}}{(2n+1)!}B_{2n+1}(1/3),$$$n\ge1$, where $B_n(\cdot)$ stands for the Bernoulli polynomial of degree~$n$.