Vorrapan Chandee: Moments of One-Level Densities for a Large Orthogonal Family..(September 11, 2025)

Moments of One-Level Densities for a Large Orthogonal Family of L-functions ~ Katz and Sarnak conjectured that the statistics of low-lying zeros of families of L-functions match with the scaling limit of eigenvalues from the random matrix theory. In this talk, I will discuss recent joint work with Yoonbok Lee and Xiannan Li on the nth-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~Q. The nth centered moments are closely related to the n-level densities of low-lying zeros of L-functions. 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). Key challenges include identifying off-diagonal main terms and resolving a combinatorial problem in matching number-theoretic results with predictions from random matrix theory. Uploaded 25-09-25 For more information go here: https://www.simonsfoundation.org/event/mps-conference-on-universal-statistics-in-number-theory/