Gaps between zeros of GL(2) $L$-functions

TL;DR

This paper investigates the distribution of zeros of GL(2) $L$-functions, proving the existence of infinitely many large and small gaps between zeros along the critical line, using advanced mean-value and pair correlation techniques.

Contribution

It establishes new bounds on the size of gaps between zeros of GL(2) $L$-functions, including both large and small gaps, by combining mean-value estimates and pair correlation results.

Findings

Infinitely many zeros have gaps at least 1.732 times the average.

Existence of infinitely many zeros with gaps less than 0.823 times the average.

Application of mean-value estimates and pair correlation results to zero spacing analysis.

Abstract

Let $L(s,f)$ be an $L$-function associated to a primitive (holomorphic or Maass) cusp form $f$ on GL(2) over $\mathbb{Q}$. Combining mean-value estimates of Montgomery and Vaughan with a method of Ramachandra, we prove a formula for the mixed second moments of derivatives of $L(1/2+it,f)$ and, via a method of Hall, use it to show that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at least $\sqrt 3 = 1.732...$ times the average spacing. Using general pair correlation results due to Murty and Perelli in conjunction with a technique of Montgomery, we also prove the existence of small gaps between zeros of any primitive $L$-function of the Selberg class. In particular, when $f$ is a primitive holomorphic cusp form on GL(2) over $\mathbb{Q}$, we prove that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at most $< 0.823$ times the average spacing.