Towards a ${\mathrm{GL}}_n$ variant of the Hoheisel phenomenon

Abstract

Let $\pi$ be a unitary cuspidal automorphic representation of ${\mathrm{GL}}_n$ over a number field, and let $\tilde{\pi }$ be contragredient to $\pi$. We prove effective upper and lower bounds of the correct order in the short interval prime number theorem for the Rankin-Selberg $L$-function $L(s,\pi \times \tilde{\pi })$, extending the work of Hoheisel and Linnik. Along the way, we prove for the first time that $L(s,\pi \times \tilde{\pi })$ has an unconditional standard zero-free region apart from a possible Landau-Siegel zero.

arXiv comment23 pages. Incorporates referee comments

References in corpus (5)

• Standard Zero-Free Regions for Rankin--Selberg L-Functions via Sieve Theory
• Weak subconvexity without a Ramanujan hypothesis
• Upper bounds on L-functions at the edge of the critical strip
• Effective log-free zero density estimates for automorphic $L$-functions and the Sato-Tate conjecture
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Cited by in corpus (3)

• A new zero-free region for Rankin-Selberg $L$-functions
• Zeros of Rankin-Selberg $L$-functions in families
• Highly Uniform Prime Number Theorems