On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges

TL;DR

This paper investigates the pair correlation of zeros of shifted auxiliary L-functions associated with non-CM newforms, providing asymptotic results and discussing a hypothesis on zero simplicity that impacts the understanding of zero distributions.

Contribution

It offers the first unconditional asymptotic analysis of zero pair correlation for these shifted auxiliary L-functions and introduces a hypothesis on zero simplicity affecting zero spacing.

Findings

Unconditional asymptotic pair correlation results for shifted auxiliary L-functions.

Introduction of a simplicity hypothesis for zeros of the auxiliary L-function.

Contrast between macroscopic and microscopic zero correlation behaviors.

Abstract

We study the pair correlation between zeros of a shifted auxiliary $ L $-function attached to a non-CM newform, the scale of which is a fixed constant. We prove an unconditional asymptotic result for the pair correlation and introduce a simplicity hypothesis for the zeros of this function, which if true means that multiple zeros of the original $ L $-function cannot be separated by the same fixed distance. Our results provide macroscopic information in contrast to the pair correlation of the original $ L $-function which is of microscopic nature.