A Riemann-Siegel formula for the Hurwitz zeta function
Let \(0<a\leq 1\) be a real number, \(\zeta(s,a)\) the Hurwitz zeta function. Let \(N\) be a natural number. The author proves that for \(s\in{\mathbb C}\) \[ \begin{multlined}\zeta(s,a)=\sum_{k=0}^{N-1}\frac{1}{(k+a)^s}+\frac{\Gamma(1-s)}{(2\pi )^{1-s}}\left\{e^{\frac{\pi i}{2}(1-s)}\sum_{k=1}^N\frac{e^{-2\pi i ka}}{k^{1-s}}+e^{-\frac{\pi i}{2}(1-s)}\sum_{k=1}^N\frac{e^{2\pi i ka}}{k^{1-s}}\right\}\\ +R_N(s,a),\end{multlined} \] with a complete estimate for the error term \(R_N(s,a)\).
scientific article; zbMATH DE number 2114293