On the Density of Low Lying Zeros of a Large Family of Automorphic $L$-functions

[2408.09050] On the Density of Low Lying Zeros of a Large Family of Automorphic $L$-functions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2408.09050 (math) [Submitted on 16 Aug 2024 ( v1 ), last revised 15 Aug 2026 (this version, v3)] Title: On the Density of Low Lying Zeros of a Large Family of Automorphic $L$-functions Authors: Timothy Cheek , Pico Gilman , Kareem Jaber , Steven J. Miller , Marie-Hélène Tomé View a PDF of the paper titled On the Density of Low Lying Zeros of a Large Family of Automorphic $L$-functions, by Timothy Cheek and 4 other authors View PDF HTML (experimental) Abstract: Under the generalized Riemann Hypothesis (GRH), Baluyot, Chandee, and Li nearly doubled the range in which the density of low lying zeros predicted by Katz and Sarnak is known to hold for a large family of automorphic $L$-functions with orthogonal symmetry. We generalize their main techniques to the study of higher centered moments of the one-level density of this family, leading to better results on the behavior near the central point. Numerous technical obstructions emerge that are not present in the one-level density. Averaging over the level of the forms and assuming GRH, we prove the density predicted by Katz and Sarnak holds for the $n$-th centered moments for test functions whose Fourier transform is compactly supported in $(-\sigma, \sigma)$ for $\sigma~=~\min\left\{3/2(n-1), 4/(2n-\mathbf{1}_{2\nmid n})\right\}$. For $n=3$, our results improve the previously best known $\sigma=2/3$ to $\sigma=3/4$. We also prove the two-level density agrees with the Katz-Sarnak density conjecture for test functions whose Fourier transform is compactly supported in $\sigma_1 = 3/2$ and $\sigma_2 = 5/6$, respectively, extending the previous best known sum of supports $\sigma_1 + \sigma_2 = 2$. This work is the first evidence of an interesting new phenomenon: by taking different test functions, we are able to extend the range in which the Katz-Sarnak density predictions hold. The techniques we develop can be applied to understanding quantities related to this family containing sums over multiple primes. Comments: 42 pages. This version fixes an error in section 3 that existed in a prior version Subjects: Number Theory (math.NT) MSC classes: 11M41 (Primary), 60B20 (Secondary) Cite as: arXiv:2408.09050 [math.NT] (or arXiv:2408.09050v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2408.09050 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Pico Gilman [ view email ] [v1] Fri, 16 Aug 2024 23:38:00 UTC (41 KB) [v2] Mon, 11 Aug 2025 23:01:25 UTC (47 KB) [v3] Sat, 15 Aug 2026 05:43:22 UTC (46 KB) Full-text links: Access Paper: View a PDF of the paper titled On the Density of Low Lying Zeros of a Large Family of Automorphic $L$-functions, by Timothy Cheek and 4 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2024-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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