Author:
Fiorilli Daniel,Parks James,Södergren Anders
Abstract
We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.
Subject
Algebra and Number Theory
Reference26 articles.
1. Applications of theL-functions ratios conjectures
2. Autocorrelation of ratios of $L$-functions
3. [BCDGL16] A. Bucur , E. Costa , C. David , J. Guerreiro and D. Lowry-Duda , Traces, high powers and one level density for families of curves over finite fields, Preprint (2016), arXiv:1610.00164.
Cited by
9 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献