# A rigidity theorem for translates of uniformly convergent Dirichlet series

## TL;DR

This paper establishes a rigidity theorem for translates of Dirichlet series in the half-plane of uniform convergence, characterizing functions approximable by such translates and relating to Bohr's equivalence theorem.

## Contribution

It provides a simple characterization of functions approximable by translates of $L$-functions in the half-plane of absolute convergence, extending the understanding of universality and rigidity.

## Key findings

- Characterization of functions approximable by translates of $L$-functions
- A rigidity theorem for Dirichlet series in the uniform convergence half-plane
- Connection to Bohr's equivalence theorem

## Abstract

It is well known that the Riemann zeta function, as well as several other $L$-functions, is universal in the strip $1/2<\sigma<1$; this is certainly not true for $\sigma>1$. Answering a question of Bombieri and Ghosh, we give a simple characterization of the analytic functions approximable by translates of $L$-functions in the half-plane of absolute convergence. Actually, this is a special case of a general rigidity theorem for translates of Dirichlet series in the half-plane of uniform convergence. Our results are closely related to Bohr's equivalence theorem.