A note on the Hurwitz zeta function (Q2724052)

Let, as usual, NEWLINE\[NEWLINE \zeta(\nu,a) = \sum_{n=0}^\infty (n+a)^{-\nu} \quad (0 < a \leq 1, \Re\nu > 1), \quad \text{Li}_\nu(z) = \sum_{k=1}^\infty z^k k^{-\nu} (\Re \nu > 1) NEWLINE\]NEWLINE denote the Hurwitz zeta-function and the polylogarithm function, respectively. For \(t \in \mathbb N\) and \(\omega = \exp(2\pi i/t)\) the authors derive the elementary identities NEWLINE\[NEWLINE \zeta\left(\nu,{s\over t}\right) = {1\over t}\sum_{r=1}^t t^\nu\text{Li}_\nu(\omega^r) \omega^{-rs}\qquad(s = 1,2,\ldots,t)\tag{1} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \text{Li}_\nu(\omega^r) = {1\over t^\nu} \sum_{s=1}^t \zeta\left(\nu,{s\over t}\right) \omega^{rs}\qquad(r = 1,2,\ldots,t).\tag{2} NEWLINE\]NEWLINE From (1) and (2) several corollaries are deduced, which involve various sums and the Bernoulli numbers. To prove (1) and (2) it suffices, by analytic continuation, to consider only \(\Re \nu > 1\). The proof of (2) is immediate from NEWLINE\[NEWLINE \text{Li}_\nu(\omega^r) = \sum_{m=0}^\infty \sum_{s=1}^t {\exp(2\pi \text{ir}(tm+s)/t)\over(tm+s)^\nu} = {1\over t^\nu}\sum_{s=1}^t\exp\left({2\pi \text{ir}s\over t}\right) \sum_{m=0}^\infty{1\over\left(m + {s\over t}\right)^\nu}. NEWLINE\]NEWLINE Substituting the expression (2) in the right-hand side of (1) we obtain (1), with the remark that this is not correctly given on p. 50, since the ``orthogonality relationship'' NEWLINE\[NEWLINE \sum_{r=1}^t\omega^{rs}\omega^{-rs} = \begin{cases} t &\text{if }r=s,\\ 0&\text{otherwise},\end{cases} NEWLINE\]NEWLINE is obviously not true (the dummy variable \(s\) in (2) is not to be confused with \(s\) in (1)).