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

Nawapan Wattanawanichkul

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arXiv · 2504.05606

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

Nawapan Wattanawanichkul

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Abstract

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

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Nawapan Wattanawanichkul. 2025-04-08. A metric approach to zero-free regions for $L$-functions. https://arxiv.org/abs/2504.05606

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