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arXiv · 1710.06834

Low-lying zeros of quadratic Dirichlet $L$-functions: A transition in the Ratios Conjecture

Daniel Fiorilli·James Parks·Anders Södergren
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Abstract

We study the $1$-level density of low-lying zeros of quadratic Dirichlet $L$-functions by applying the $L$-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the lower order terms when the support of the Fourier transform of the corresponding test function reaches the point $1$. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case.

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Daniel Fiorilli, James Parks, Anders Södergren. 2017-10-18. Low-lying zeros of quadratic Dirichlet $L$-functions: A transition in the Ratios Conjecture. https://arxiv.org/abs/1710.06834

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