A note on Bohr’s theorem for Beurling integer systems

Abstract

Given a sequence of frequencies , a corresponding generalized Dirichlet series is of the form . We are interested in multiplicatively generated systems, where each number arises as a finite product of some given numbers , , referred to as Beurling primes. In the classical case, where , Bohr’s theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane , then it actually converges uniformly in every half-plane , . We prove, under very mild conditions, that given a sequence of Beurling primes, a small perturbation yields another sequence of primes such that the corresponding Beurling integers satisfy Bohr’s condition, and therefore the theorem. Applying our technique in conjunction with a probabilistic method, we find a system of Beurling primes for which both Bohr’s theorem and the Riemann hypothesis are valid. This provides a counterexample to a conjecture of H. Helson concerning outer functions in Hardy spaces of generalized Dirichlet series.