On Hypothesis H of Rudnick and Sarnak

TL;DR

This paper proves Hypothesis H for GL_n over any number field using a new analytic method, leading to significant advances in understanding automorphic L-functions, multiplicity problems, and orthogonality conjectures.

Contribution

It introduces a novel analytic approach employing a power sieve and iterative techniques to extend results to all n, surpassing previous limitations.

Findings

Establishes GUE statistics for automorphic L-function zeros.

Provides the first effective polynomial bound for strong multiplicity one.

Resolves the Selberg orthogonality conjecture with improved error terms.

Abstract

We prove Hypothesis H in full generality for ${\rm GL}_n$ over any number field. This result is a consequence of our stronger effective bound on Euler products involving Rankin--Selberg coefficients at prime ideal powers. The proof rests on a new analytic method, which employs a power sieve over number fields and an iterative argument to bypass the functoriality barrier that had restricted prior results to $n\leq 4$. As applications, we unconditionally establish the GUE statistics for automorphic $L$-function zeros, provide the first effective polynomial bound for the strong multiplicity one problem for coefficients, and resolve the Selberg orthogonality conjecture with stronger error terms.