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

TL;DR

This paper investigates when two functions in the extended Selberg class are identical based on the zeros of their difference with a moving target, introducing a new method inspired by complex dynamics.

Contribution

It establishes conditions under which two Selberg class functions are identical using a novel approach based on transcendental directions.

Findings

Proves L_1 ≡ L_2 if L_1 - h and L_2 - h share zeros with multiplicities.

Constructs examples showing the growth order assumption on h is necessary.

Develops a new strategy based on complex dynamical systems concepts.

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.