A zero-density estimate for $L$-functions associated with $\rm GL(3)$ Hecke--Maass cusp forms

TL;DR

This paper derives an asymptotic formula for the second moment of $L$-functions linked to $ m SL(3, ext{Z})$ Hecke--Maass cusp forms and uses it to estimate the density of their zeros, advancing understanding of their distribution.

Contribution

It provides a new asymptotic formula for the twisted second moment and a weighted zero-density estimate for $ m SL(3, ext{Z})$ $L$-functions, which was previously unexplored.

Findings

Established an asymptotic formula for the second moment of $L$-functions.

Derived a weighted zero-density estimate in the spectral aspect.

Potential applications in related problems involving $ m SL(3)$ automorphic forms.

Abstract

In this paper, we establish an asymptotic formula for the twisted second moment of $L$-functions associated with Hecke--Maass cusp forms for $\rm SL(3,\mathbb{Z})$, and further deduce a weighted zero-density estimate for these $L$-functions in the spectral aspect which may have important applications in other problems.