# $n$-level density of the low-lying zeros of primitive Dirichlet $L$-functions

## TL;DR

This paper confirms that the low-lying zeros of primitive Dirichlet L-functions statistically match the eigenvalue distribution of the random unitary ensemble, supporting the Katz-Sarnak conjecture through advanced analytical methods.

## Contribution

It provides a rigorous confirmation of the Katz-Sarnak conjecture for primitive Dirichlet L-functions, including off-diagonal contributions, using the asymptotic large sieve and random matrix theory formulas.

## Key findings

- Low-lying zeros match random matrix predictions
- Includes off-diagonal contributions in the analysis
- Uses asymptotic large sieve for density estimation

## Abstract

Katz and Sarnak conjectured that the statistics of low-lying zeros of various family of $L$-functions matched with the scaling limit of eigenvalues from the random matrix theory. In this paper we confirm this statistic for a family of primitive Dirichlet $L$-functions matches up with corresponding statistic in the random unitary ensemble, in a range that includes the off-diagonal contribution. To estimate the $n$-level density of zeros of the $L$-functions, we use the asymptotic large sieve method developed by Conrey, Iwaniec and Soundararajan. For the random matrix side, a formula from Conrey and Snaith allows us to solve the matchup problem.