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

Abstract

Let $π$ and $π'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. Let $\mathfrak{C}_π$ be the analytic conductor of $π$. We develop a new approach to zero-free regions for $L$-functions via lower bounds for power sums, proving for all $\varepsilon>0$ the existence of ineffective constants $c=c_{n,F,\varepsilon}>0$ and $c'=c'_{n,F,π',\varepsilon}>0$ such that the standard $L$-function $L(s,π)$ 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 $L(s,π\timesπ')$ 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.