On zeros of the Lerch zeta-function (Q2707563)

The Lerch zeta-function is given by \(L(\lambda, \alpha, s)= \sum_{n=0}^\infty \exp(2\pi i\lambda n) (n+ \alpha)^{-s}\) or its analytic continuation. In the paper under review the authors study the zero distribution of \(L(\lambda, \alpha, s)\) as an entire function in \(s\) with fixed parameters \(0< \lambda < 1\); \(0< \alpha \leq 1\). They prove zero-free regions in the left and in the right half plane; more exactly: There are no zeros of \(L(\lambda, \alpha,s)\) in \(\operatorname {Re}s\geq 1+\alpha\), and all zeros in \(\operatorname {Re}s <0\) lie close to a straight line, depending on the parameters \(\lambda, \alpha\). Further, they prove a Riemann-von Mangoldt type formula for the number of zeros in \(0<\operatorname {Im} s< T\) and \(-T<\operatorname {Im}s<0\).NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].