A rigidity theorem for translates of uniformly convergent Dirichlet series

It is well known that the Riemann zeta function, as well as several other $L$-functions, is universal in the strip $1/2<1$; 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.