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

Abstract

For integers $m,m^{\prime }\ge 1$, let $\pi$ and ${\pi }^{\prime }$ be cuspidal automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(m^{\prime })$, respectively. We present a new proof of zero-free regions for $L(s,\pi )$ and for $L(s,\pi \times {\pi }^{\prime })$ under the assumption that $\pi ,{\pi }^{\prime }$ or $L(s,\pi \times {\pi }^{\prime })$ 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.

25 pages; added references and an outline of the proof in the introduction; fixed typos and minor calculation errors in Theorem 1.1