Zeros of linear twists of $L$-functions outside the critical strip

TL;DR

This paper proves that the Lerch zeta function has infinitely many zeros outside the critical strip, specifically for real parts slightly greater than 1, when certain parameters are irrational and rational, respectively.

Contribution

It establishes the existence of infinitely many zeros of the Lerch zeta function outside the critical strip for the first time.

Findings

Lerch zeta function has infinitely many zeros for 1<σ<1+η when λ is irrational and α is rational.

Zeros are shown to exist for any η>0, extending the understanding of zero distribution.

Settles the question of zeros of Lerch zeta functions for σ>1.

Abstract

In this note we investigate the existence of zeros of linear twists of $L$-functions outside of the critical strip. In particular, we show that the Lerch zeta function $L(\lambda,\alpha,s)$ has infinitely many zeros for $1<\sigma<1+\eta$, for any $\eta>0$, when $\lambda$ is irrational and $\alpha$ is rational. This settles the question on the existence of zeros of the Lerch zeta functions for $\sigma>1$.