Standard Zero-Free Regions for Rankin--Selberg L-Functions via Sieve Theory

Abstract

We give a simple proof of a standard zero-free region in the $t$-aspect for the Rankin--Selberg $L$-function $L(s,\pi \times \tilde{\pi })$ for any unitary cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_n({\mathbb{A}}_F)$ that is tempered at every nonarchimedean place outside a set of Dirichlet density zero.

16 pages. With an appendix by Farrell Brumley

References in corpus (2)

• Primes and prime ideals in short intervals
• Effective Lower Bounds for L(1,χ) via Eisenstein Series

Cited by in corpus (14)

• Weak subconvexity without a Ramanujan hypothesis
• Effective forms of the Sato--Tate conjecture
• The Weyl bound for triple product L-functions
• A Bombieri-Vinogradov theorem for higher rank groups
• An unconditional $\mathrm{GL}(n)$ large sieve
• Effective log-free zero density estimates for automorphic $L$-functions and the Sato-Tate conjecture
• Towards a ${\mathrm{GL}}_n$ variant of the Hoheisel phenomenon
• Zeros of Rankin-Selberg $L$-functions in families
• Patterns of primes in the Sato-Tate conjecture
• A new zero-free region for Rankin-Selberg $L$-functions
• New variants of arithmetic quantum ergodicity
• Highly Uniform Prime Number Theorems
• Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions
• An Asymptotic Orthogonality Relation for $\mathrm{GL}(n,\mathbb{R})$