Lower Bounds for Rankin-Selberg $L$-functions on the Edge of the Critical Strip

TL;DR

This paper establishes new lower bounds and zero-free regions for Rankin-Selberg $L$-functions on the line $ ext{Re } s=1$, enhancing understanding of their behavior near the edge of the critical strip in the $t$-aspect.

Contribution

It provides the first explicit lower bounds and zero-free regions for Rankin-Selberg $L$-functions at the edge of the critical strip, extending previous results.

Findings

Derived explicit lower bounds for $L(s,  imes \u0012)$ on $ ext{Re } s=1$

Identified zero-free regions for these $L$-functions near the critical line

Enhanced understanding of the $t$-aspect behavior of Rankin-Selberg $L$-functions

Abstract

Let $F$ be a number field, and let $\pi_1$ and $\pi_2$ be distinct unitary cuspidal automorphic representations of $\operatorname{GL}_{n_1}(\mathbb{A}_F)$ and $\operatorname{GL}_{n_2}(\mathbb{A}_F)$ respectively. In this paper, we derive new lower bounds for the Rankin-Selberg $L$-function $L(s, \pi_1 \times \widetilde{\pi}_2)$ along the edge $\Re s = 1$ of the critical strip in the $t$-aspect. The corresponding zero-free region for $L(s, \pi_1 \times \widetilde{\pi}_2)$ is also determined.