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arXiv · 2510.07647

The $n^{th}$ centered moments of a large orthogonal family of automorphic $L$-functions

Vorrapan Chandee·Yoonbok Lee·Xiannan Li
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Abstract

We obtain the $n$th centered moments of one level densities of a large orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q\sim Q$. We verify the Katz-Sarnak conjecture for these statistics, in the range where the sum of the supports of the Fourier transforms of test functions lies in $(-4, 4)$. In so doing, we need to understand certain phantom oversized terms, which allow us to extract the right off-diagonal contributions. We further need to resolve the combinatorial problem that arises when matching our main terms with random matrix predictions.

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Vorrapan Chandee, Yoonbok Lee, Xiannan Li. 2025-10-09. The $n^{th}$ centered moments of a large orthogonal family of automorphic $L$-functions. https://arxiv.org/abs/2510.07647

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