A note on Bohr's theorem for Beurling integer systems

Abstract

Given a sequence of frequencies $\{\lambda_n\}_{n\geq1}$, a corresponding generalized Dirichlet series is of the form $f(s)=\sum_{n\geq 1}a_ne^{-\lambda_ns}$. We are interested in multiplicatively generated systems, where each number $e^{\lambda_n}$ arises as a finite product of some given numbers $\{q_n\}_{n\geq 1}$, $1 < q_n \to \infty$, referred to as Beurling primes. In the classical case, where $\lambda_n = \log n$, Bohr's theorem holds: if $f$ converges somewhere and has an analytic extension which is bounded in a half-plane $\{\Re s> \theta\}$, then it actually converges uniformly in every half-plane $\{\Re s> \theta+\varepsilon\}$, $\varepsilon>0$. 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.