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arXiv · 2507.20653

On Hypothesis H of Rudnick and Sarnak

Yujiao Jiang
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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.

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Yujiao Jiang. 2025-07-28. On Hypothesis H of Rudnick and Sarnak. https://arxiv.org/abs/2507.20653

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