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

TL;DR

This work extends Bohr–Landau type zero-density results to a broad class of analytic functions $F(s,\bm{\pi})$ formed as polynomials in derivatives of automorphic $L$-functions. By constructing a robust analytic framework, the authors derive an explicit asymptotic for the nontrivial zeros up to height $T$, with main term $\alpha_1 T\log T + \alpha_2 T$ where $\alpha_1$ and $\alpha_2$ are given in terms of rank, derivative, and conductor data. They establish a Hadamard-type factorization for $F$ (up to a finite pole) and a precise zero-free region, enabling a complete zero-count. Under a strong second-moment bound for the component $L$-functions, they prove that almost all nontrivial zeros lie within a small distance of the critical line $\mathrm{Re}(s)=\tfrac{1}{2}$, and they connect these results to classical problems on $a$-points and truncated Taylor expansions. The paper unifies and extends classical results for $\zeta(s)$ and its derivatives to the automorphic setting, providing a versatile toolkit for the zero-distribution of complex combinations of $L$-functions.

Abstract

Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbolπ)$ be a polynomial in the derivatives of $L$-functions associated with representations $π_u \in \cup_{m=1}^{\infty} \mathfrak{F}_m$. We establish an asymptotic formula for the number of nontrivial zeros of $F(s,\boldsymbolπ)$ 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 $F(s,\boldsymbolπ)$ lie near the critical line $\operatorname{Re}(s)=1/2$.

Figures (1)

• Figure 1: The rectangular contour and the encompassing quarter-circular arc

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof