A note on Bohr’s theorem for Beurling integer systems

Given a sequence of frequencies { λ n } n ≥ 1 , a corresponding generalized Dirichlet series is of the form f ( s ) = ∑ n ≥ 1 a n e - λ n s . We are interested in multiplicatively generated systems, where each number e λ n arises as a finite product of some given numbers { q n } n ≥ 1 , 1 < q n → ∞ , referred to as Beurling primes. In the classical case, where λ n = log n , Bohr’s theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane { ℜ s > θ } , then it actually converges uniformly in every half-plane { ℜ s > θ + ε } , ε > 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.