A note on the Hurwitz zeta function

Abstract

We show that the Hurwitz zeta function and polylogarithm, $\zeta (\nu ,a)$ and ${\mathrm{Li}}_{\nu }(z)$, form a discrete Fourier transform pair for $\mathrm{\Re }\nu >1$. 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, $$2\zeta (2n+1,1/3)=(3^{2n+1}-1)\zeta (2n+1)+(-1{)}^{n-1}{3}^{2n}\sqrt{3}{\displaystyle \frac{(2\pi {)}^{2n+1}}{(2n+1)!}}{B}_{2n+1}(1/3),$$$n\ge 1$, where $B_n(\cdot )$ stands for the Bernoulli polynomial of degree~$n$.