On uniqueness of functions in the extended Selberg class with moving targets

TL;DR

The paper studies the uniqueness of L-functions in the extended Selberg class under moving-target perturbations $h$. It develops a novel approach based on transcendental directions from complex dynamics to compare two L-functions $L_1$ and $L_2$ via the quotient $\varphi=(L_1-h)/(L_2-h)$ and shows that $L_1\\equiv L_2$ whenever $L_1-h$ and $L_2-h$ share zeros counting multiplicities, under precise growth/order assumptions on $h$ (e.g., $\rho(h)\\neq 1$ for entire $h$, and suitable conditions for meromorphic $h$). The results extend and sharpen prior moving-target theorems by Cardwell–Ye and Hu–Li, and the authors provide examples to demonstrate the necessity of the assumptions. This work bridges Nevanlinna theory and values-distribution techniques with L-function uniqueness, introducing a method that may be of independent interest for moving-target problems in analytic number theory and complex dynamics.

Abstract

We study the question of when two functions L_1,L_2 in the extended Selberg class are identical in terms of the zeros of L_i-h(i=1,2). Here, the meromorphic function h is called moving target. With the assumption on the growth order of h, we prove that L_1\equiv L_2 if L_1-h and L_2-h have the same zeros counting multiplicities. Moreover, we also construct some examples to show that the assumption is necessary. Compared with the known methods in the literature of this area, we developed a new strategy which is based on the transcendental directions first proposed in the study of distribution of Julia set in complex dynamical system. This may be of independent interest.

Theorems & Definitions (18)

• Theorem A
• Theorem B
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Example 1
• Example 2
• Example 3
• Lemma 1