Zeros of Polynomials in Derivatives of Automorphic $L$-functions

TL;DR

This paper studies the distribution of zeros of polynomials formed from derivatives of automorphic $L$-functions, providing asymptotic counts and showing most zeros lie near the critical line under certain conditions.

Contribution

It establishes an explicit asymptotic formula for zeros of such polynomials and characterizes their typical location relative to the critical line.

Findings

Asymptotic formula for the number of zeros with imaginary part up to T.

Main term expressed in terms of degrees, ranks, conductors, and derivatives.

Most zeros are near the critical line $ ext{Re}(s)=1/2$ under certain conditions.

Abstract

Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbol{\pi})$ be a polynomial in the derivatives of $L$-functions associated with representations $\pi_u \in \cup_{m=1}^{\infty} \mathfrak{F}_m$. We establish an asymptotic formula for the number of nontrivial zeros of $F(s,\boldsymbol{\pi})$ with $0 < \operatorname{Im}(s) < T$. We explicitly determine the main term of this formula in terms of the degrees, the ranks, the arithmetic conductors, and the orders of differentiation of the component $L$-functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of $F(s,\boldsymbol{\pi})$ lie near the critical line $\operatorname{Re}(s)=1/2$.