Low-lying zeros of cubic Dirichlet $L$-functions and the Ratios Conjecture

TL;DR

This paper computes the one-level density for cubic Dirichlet L-functions within a specific support range and confirms the Ratios conjecture's prediction aligns with the empirical results.

Contribution

It provides the first explicit computation of the one-level density for cubic Dirichlet L-functions and verifies the Ratios conjecture's accuracy for this family.

Findings

Confirmed the Ratios conjecture prediction for the family

Computed the one-level density for cubic Dirichlet L-functions

Supported the conjecture within the Fourier transform support (-1,1)

Abstract

We compute the one-level density for the family of cubic Dirichlet $L$-functions when the support of the Fourier transform of a test function is in $(-1,1)$. We also establish the Ratios conjecture prediction for the one-level density for this family, and confirm that it agrees with the one-level density we obtain.