An approximate functional equation for the Lerch zeta-function. (Q1889561)

Let \(0<\alpha\), \(\alpha \leq 1\). The Lerch zeta-function \(L(\lambda,\alpha,s)\), \(s=\sigma+it\), for \(\sigma>1\), is defined by \[ L(\lambda,\alpha,s)=\sum_{n=0}^{\infty}\frac{e^{2 \pi i \lambda n}}{(n+\alpha)^s}. \] If \(0<\lambda<1\), then the function \(L(\lambda,\alpha,s)\) is analytically continuable to an entire function, while for \(\lambda=1\) it reduces to the Hurwitz zeta-function. In the paper, the following approximate functional equation for \(L(\lambda,\alpha,s)\) is proved. Let \(0\leq \sigma \leq 1\), \(t \geq t_0 >1\), \(y=\big(\frac{t}{2 \pi}\big)^{1/2}\), \(q=[y]\), \(m=[y-\alpha]\) and \(\beta=q-m\). Then \[ \begin{multlined} L(\lambda,\alpha,s)= \sum_{k=0}^{m}\frac{e^{2 \pi i \lambda k}}{(k+\alpha)^s}+\bigg(\frac{2 \pi}{t}\bigg)^{\sigma-\frac{1}{2}+it}e^{it +\frac{\pi i}{4}-2 \pi i \{\lambda\}\alpha} \sum_{n=0}^{q}\frac{e^{- 2 \pi i \alpha n}}{(n+\alpha)^{1-s}} \\ +\bigg(\frac{2 \pi}{t}\bigg)^{\frac{\sigma}{2}} e^{\pi i f(\lambda,\alpha,\sigma,t)} \psi(2y-2q+\beta-\{\lambda\}-\alpha)+O(t^{\frac{\sigma-2}{2}}).\end{multlined} \] Here \[ f(\lambda,\alpha,\sigma,t)=-\frac{t}{2 \pi}\log \frac{t}{2 \pi e}-\frac{7}{8}+\frac{1}{2}(\alpha^2-\{\lambda\}^2)-\alpha \beta + 2 y (\beta+\{\lambda\}-\alpha)-\frac{1}{2}(q+m)-\{\lambda\}(\beta+\alpha), \] and \[ \psi(a)=\frac{\cos (\pi(a^2/2-a-1/8))}{\cos (\pi a)}. \] The latter result generalizes an approximate functional equation in the case \(\sigma=\frac{1}{2}\) for the Hurwitz zeta-function obtained by \textit{V. V. Rane} [J. Lond. Math. Soc. (2) 21, 203--215 (1980; Zbl 0422.10032)].