A zero density estimate and fractional imaginary parts of zeros for $\mathrm{GL}_2$ $L$-functions

TL;DR

This paper extends zero density estimates to $ ext{GL}_2$ $L$-functions and investigates the distribution of fractional parts of their zeros' imaginary parts, providing new insights into their zero distribution.

Contribution

It establishes a zero density estimate for $ ext{GL}_2$ $L$-functions and analyzes the distribution of fractional parts of zeros' imaginary components.

Findings

Zero density estimate analogous to Selberg's for $ ext{GL}_2$ $L$-functions

Distribution results for fractional parts of zeros' imaginary parts

Insights into the zero distribution of $ ext{GL}_2$ $L$-functions

Abstract

We prove an analogue of Selberg's zero density estimate for $\zeta(s)$ that holds for any $\mathrm{GL}_2$ $L$-function. We use this estimate to study the distribution of the vector of fractional parts of $\gamma\mathbf{\alpha}$, where $\mathbf{\alpha}\in\mathbb{R}^n$ is fixed and $\gamma$ varies over the imaginary parts of the nontrivial zeros of a $\mathrm{GL}_2$ $L$-function.