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

TL;DR

This paper investigates the distribution of low-lying zeros of quadratic Dirichlet L-functions using the Ratios Conjecture, revealing a transition in the main term at a critical support point, consistent with heuristic predictions.

Contribution

It demonstrates a transition in the 1-level density of zeros at support 1, aligning the Ratios Conjecture with the Katz-Sarnak heuristic and previous GRH-based results.

Findings

Identifies a transition in the main term at support 1.

Results match previous GRH-based findings.

Analogous to Rudnick's results in the function field case.

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.