A new zero-free region for Rankin-Selberg $L$-functions

TL;DR

This work establishes a new Siegel-type zero-free region for all GL1 twists of Rankin–Selberg L-functions $L(s,π×π')$, extending Siegel’s classical results to the automorphic setting. The authors develop a novel method that leverages the GL1 group structure and a nonnegative-coefficient auxiliary L-function to derive lower bounds at $s=1$ from hypothetical zeros near the line Re(s)=1, then propagate these bounds across twists to obtain a zero-free region with ineffective constants. The approach yields two principal applications: (i) an unconditional Siegel–Walfisz-type theorem for the Dirichlet coefficients of $-L'/L(s,π×π')$, and (ii) an enlarged region of holomorphy and nonvanishing for twisted symmetric-power L-functions $L(s,π, ext{Sym}^n⊗χ)$ for $n≤8$, with automorphy and Clebsch–Gordan tools handling the higher cases. Consequently, the results enable a Leiter of consequences, including a PNT-type statement in ray class progressions and deeper control over symmetric-power twists in the automorphic setting, advancing the understanding of L-functions in the Langlands program.

Abstract

Let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Siegel's celebrated work on Dirichlet $L$-functions. As an application, we prove the first unconditional Siegel-Walfisz theorem for the Dirichlet coefficients of $-L'(s,π\timesπ')/L(s,π\timesπ')$. Also, for $n\leq 8$, we extend the region of holomorphy and nonvanishing for the twisted symmetric power $L$-functions $L(s,π,\mathrm{Sym}^n\otimesχ)$ of any cuspidal automorphic representation of $\mathrm{GL}(2)$.

Theorems & Definitions (24)

• Theorem 1.1
• Remark
• Remark
• Theorem 2.1
• Remark
• Theorem 2.2
• Lemma 3.1: HoffsteinRamakrishnan
• Lemma 3.2
• proof
• Lemma 3.3