A variant of the Linnik-Sprindzuk theorem for simple zeros of Dirichlet L-functions

Abstract

For a primitive Dirichlet character $X$, a new hypothesis $RH_{sim}^\dagger[X]$ is introduced, which asserts that (1) all simple zeros of $L(s,X)$ in the critical strip are located on the critical line, and (2) these zeros satisfy some specific conditions on their vertical distribution. We show that $RH_{sim}^\dagger[X]$ (for any $X$) is a consequence of the generalized Riemann hypothesis. Assuming only the generalized Lindel\"of hypothesis, we show that if $RH_{sim}^\dagger[X]$ holds for one primitive character $X$, then it holds for every such $X$. If this occurs, then for every character $\chi$ (primitive or not), all simple zeros of $L(s,\chi)$ in the critical strip are located on the critical line. In particular, Siegel zeros cannot exist in this situation.