Low-lying Zeros of Number Field $L$-functions

TL;DR

This paper extends the understanding of zero distributions of L-functions by analyzing the 1-level density for general number field families, confirming predictions from random matrix theory and revealing arithmetic influences in lower order terms.

Contribution

It generalizes previous results on 1-level density to broader number field families and derives the first lower order term showing arithmetic effects.

Findings

Main term matches random matrix theory predictions

Lower order term reveals arithmetic dependence

Results apply to general sequences of number fields

Abstract

One of the most important statistics in studying the zeros of L-functions is the 1-level density, which measures the concentration of zeros near the central point. Fouvry and Iwaniec [FI] proved that the 1-level density for L-functions attached to imaginary quadratic fields agrees with results predicted by random matrix theory. In this paper, we show a similar agreement with random matrix theory occurring in more general sequences of number fields. We first show that the main term agrees with random matrix theory, and similar to all other families studied to date, is independent of the arithmetic of the fields. We then derive the first lower order term of the 1-level density, and see the arithmetic enter.