Power sums and Siegel-type zero-free regions for L-functions

Abstract

Let $\pi$ and ${\pi }^{\prime }$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n^{\prime })$ over a number field $F$. Let ${\mathfrak{C}}_{\pi }$ be the analytic conductor of $\pi$. We develop a new approach to zero-free regions for $L$-functions via lower bounds for power sums, proving for all $\epsilon >0$ the existence of ineffective constants $c=c_{n,F,\epsilon }>0$ and $c^{\prime }=c_{n,F,{\pi }^{\prime },\epsilon }^{\prime }>0$ such that the standard $L$-function $L(s,\pi )$ satisfies \[ |L(σ+it,π)|\geq c(\mathfrak{C}_π(|t|+3))^{-\varepsilon},\qquad σ\geq 1-c(\mathfrak{C}_π(|t|+3))^{-\varepsilon} \] and the Rankin-Selberg $L$-function $L(s,\pi \times {\pi }^{\prime })$ satisfies \[ |L(σ+it,π\timesπ')|\geq c'(\mathfrak{C}_π(|t|+3))^{-\varepsilon},\qquad σ\geq 1-c'(\mathfrak{C}_π(|t|+3))^{-\varepsilon}. \] Applications include improvements to the prime number theorems for these $L$-functions and new generalizations of the Brauer-Siegel theorem.

arXiv comment21 pages