Upper bounds on gaps between zeros of $L$-functions

TL;DR

This paper establishes two unconditional upper bounds on the gaps between zeros of general $L$-functions, extending prior work on the Riemann zeta-function and Dirichlet $L$-functions.

Contribution

It introduces new bounds applicable to a broad class of $L$-functions, comparing the effectiveness of different methods depending on the function's degree and conductor.

Findings

Hall and Hayman's method is sharper for small degree $L$-functions.

Siegel's method performs better for larger degree or conductor.

The results unify bounds across different types of $L$-functions.

Abstract

We prove two unconditional upper bounds on the gaps between ordinates of consecutive non-trivial zeros of a general $L$-function $L(s)$. This extends previous work of Hall and Hayman (2000) on the Riemann zeta-function and work of Siegel (1945) on Dirichlet $L$-functions. Interestingly, we observe that while Hall and Hayman's method gives a sharper estimate when the degree of $L(s)$ is sufficiently small compared to the analytic conductor, Siegel's method does better in the other regime.