On Hypothesis H of Rudnick and Sarnak
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912505185435648 |
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| author | Jiang, Yujiao |
| author_facet | Jiang, Yujiao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20653 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Hypothesis H of Rudnick and Sarnak Jiang, Yujiao Number Theory 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. |
| title | On Hypothesis H of Rudnick and Sarnak |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.20653 |