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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2508.14888 |
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| _version_ | 1866916948597538816 |
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| author | Pascadi, Alexandru Thorner, Jesse |
| author_facet | Pascadi, Alexandru Thorner, Jesse |
| contents | Let $\mathfrak{F}_n$ be the set of unitary cuspidal automorphic representations of $\mathrm{GL}_n$ over a number field $F$, and let $S\subseteq\mathfrak{F}_n$ be an arbitrary finite subset. Given $π_0\in\mathfrak{F}_{n_0}$, we establish large sieve inequalities for the families $\{L(s,π)\colon π\in S\}$ and $\{L(s,π\timesπ_0)\colon π\in S\}$ that, unlike previous results, are independent of progress towards the generalized Ramanujan conjecture, and simultaneously handle the Dirichlet coefficients of $L$, $L^{-1}$, and $\log L$. We also give the first such result that improves upon the trivial bound for short sums. We present several applications, including:
(1) the strongest bound for $\sum_{π\in S}|L(\frac{1}{2},π)|^2$ that holds for arbitrary $S$,
(2) significant improvements to zero density estimates for families of automorphic and Rankin--Selberg $L$-functions, counting violations to the generalized Riemann hypothesis near $\mathrm{Re}(s)=1$,
(3) the removal of all unproven hypotheses in the conditional log-free zero density estimate for families of Rankin--Selberg $L$-functions proved by Brumley, Thorner, and Zaman, and
(4) an improvement of the density theorem for non-archimedean Langlands parameters due to Lichtman and Pascadi, counting violations to the generalized Ramanujan conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14888 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large sieves for $\mathrm{GL}_n$ and applications Pascadi, Alexandru Thorner, Jesse Number Theory Let $\mathfrak{F}_n$ be the set of unitary cuspidal automorphic representations of $\mathrm{GL}_n$ over a number field $F$, and let $S\subseteq\mathfrak{F}_n$ be an arbitrary finite subset. Given $π_0\in\mathfrak{F}_{n_0}$, we establish large sieve inequalities for the families $\{L(s,π)\colon π\in S\}$ and $\{L(s,π\timesπ_0)\colon π\in S\}$ that, unlike previous results, are independent of progress towards the generalized Ramanujan conjecture, and simultaneously handle the Dirichlet coefficients of $L$, $L^{-1}$, and $\log L$. We also give the first such result that improves upon the trivial bound for short sums. We present several applications, including: (1) the strongest bound for $\sum_{π\in S}|L(\frac{1}{2},π)|^2$ that holds for arbitrary $S$, (2) significant improvements to zero density estimates for families of automorphic and Rankin--Selberg $L$-functions, counting violations to the generalized Riemann hypothesis near $\mathrm{Re}(s)=1$, (3) the removal of all unproven hypotheses in the conditional log-free zero density estimate for families of Rankin--Selberg $L$-functions proved by Brumley, Thorner, and Zaman, and (4) an improvement of the density theorem for non-archimedean Langlands parameters due to Lichtman and Pascadi, counting violations to the generalized Ramanujan conjecture. |
| title | Large sieves for $\mathrm{GL}_n$ and applications |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.14888 |