A metric approach to zero-free regions for $L$-functions

Abstract

For integers $m, m' \ge 1$, let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(m')$, respectively. We present a new proof of zero-free regions for $L(s, π)$ and for $L(s, π\times π')$ under the assumption that $π, π'$ or $L(s,π\times π')$ is self-dual. Our approach builds on ideas of "pretentious" multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic representations due to Lichtman and Pascadi.