Twists of Lerch zeta-functions
This paper is on some basic properties of twists of Lerch zeta-functions defined as \[ L(\lambda, \alpha, s, \chi, Q) = \sum_{n=0}^{\infty}{\chi(n+Q)e^{2\pi i\lambda n}\over (n+\alpha)^{s}} \quad (\Re s > 1), \] where \(0 < \alpha\leq 1\), \(\lambda\in \mathbb R\), \(Q\in \mathbb Z\) and \(\chi\) is a Dirichlet character to the modulus \(q\). For \(\chi\equiv 1\) this is the Lerch zeta-function, which in turn reduces to the Hurwitz zeta-function if \(\lambda\in \mathbb Z\). In this paper meromorphic continuation and functional equations of the twists of Lerch zeta-functions are determined. The authors state that after the preparation of this paper they learned about the works by \textit{B. C. Berndt} [J. Number Theory 7, 413-445 (1975; Zbl 0316.10023)] and \textit{B. C. Berndt} and \textit{L. Schoenfeld} [Acta Arith. 28, 23-68 (1975; Zbl 0268.10008)] which contained similar results. The analogue of the Riemann-von Mangoldt formula for the number of zeros with imaginary parts in \([-T,T]\) is also proved. Moreover it is shown, using Voronin's theorem on the universality of Dirichlet \(L\)-functions, that the analogue of the Riemann hypothesis does not hold for certain twists.
scientific article; zbMATH DE number 1912573