Identification of L-functions in the extended Selberg class by preimages of finite sets

TL;DR

This paper advances the uniqueness theory of L-functions in the extended Selberg class by removing the need for shared functional equations and establishing new criteria based on finite value sets.

Contribution

It significantly improves previous results by weakening hypotheses and proves that polynomials with distinct zeros are strong uniqueness polynomials for L-functions.

Findings

Removing the shared functional equation assumption still guarantees L-function equality.

Sharing a finite set of complex values with multiplicities suffices for uniqueness.

Polynomials with distinct zeros are strong uniqueness polynomials for L-functions.

Abstract

In 2023, Li, Du, Yi proved a uniqueness theorem for L functions in the extended Selberg class under the assumptions of positive degree, a shared functional equation, and the sharing of three complex values. This was later strengthened by the present authors, who showed that sharing an arbitrary finite set of complex values, counted with multiplicities, still forces equality of the two L functions, again under the assumption that they satisfy the same functional equation. In this paper, we significantly improve all these results. We completely remove the requirement that the two L functions satisfy the same functional equation, yet we still obtain the same strong uniqueness conclusion under far weaker hypotheses. As a major consequence, we prove that every polynomial with distinct zeros is a strong uniqueness polynomial for L functions.