Tatuzawa's theorem for Rankin-Selberg $L$-functions
Gergely Harcos, Jesse Thorner
TL;DR
This work addresses zero-free regions for GL1 twists of Rankin–Selberg L-functions associated to unitary cuspidal automorphic representations, generalizing Tatuzawa’s refinement of Siegel-type results. It develops a uniform, twist-sensitive framework based on a refined Goldfeld–Hoffstein–Lieman approach, isobaric sums, and residue calculus to obtain effective zero-free regions without assuming unproven hypotheses. The main result yields an effectively computable region in which $L(s,\pi\times\pi')$ has at most one zero (which is simple), with a single possible exceptional twist when $\pi\otimes\chi^2=\pi$, and extends to all GL1 twists of $L(s,\pi\times\pi')$. These Tatuzawa-type bounds tighten the understanding of zeros near $s=1$ for automorphic L-functions and have potential implications for nonvanishing results and arithmetic applications tied to special values, conductors, and twists.
Abstract
Let $π$ and $π'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. As a corollary, we show that for all $\varepsilon>0$, there exists an effectively computable constant $c>0$ depending only on $(n,n',[F:\mathbb{Q}],\varepsilon)$ such that $L(s,π\timesπ')$ has at most one zero (necessarily simple) in the region \[ \mathrm{Re}(s)\geq 1-c/(C(π)C(π')(|\mathrm{Im}(s)|+1))^{\varepsilon}, \] where $C(π)$ and $C(π')$ are the analytic conductors. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,π\times\widetildeπ)$ by an idele class character $χ$ apart from a possible single exceptional zero (necessarily real and simple) that can occur only when $π\otimesχ^2=π$. This extends earlier work of Humphries and Thorner.