One Level Density for Cubic Galois Number Fields

TL;DR

This paper proves that the zeros of L-functions from cubic Galois number fields follow the one level density distribution associated with unitary matrices, confirming a specific case of Katz and Sarnak's prediction.

Contribution

It establishes the one level density for L-functions of cubic Galois number fields as belonging to the unitary class, providing evidence for Katz and Sarnak's conjecture in this case.

Findings

Zeros follow the unitary matrix distribution

Supports Katz and Sarnak's classification for this family

Advances understanding of L-functions in number theory

Abstract

Katz and Sarnak predicted that the one level density of the zeros of a family of $L$-functions would fall into one of five categories. In this paper, we show that the one level density for $L$-functions attached to cubic Galois number fields falls into the category associated with unitary matrices.