A note on Bohr’s theorem for Beurling integer systems

Given a sequence of frequencies {.n} n=1, a corresponding generalized Dirichlet series is of the form f (s) = n=1 ane-.n s. We are interested in multiplicatively generated systems, where each number e.n arises as a finite product of some given numbers {qn} n=1, 1 < qn. 8, referred to as Beurling primes. In the classical case, where.n = log n, Bohr's theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane {s >.}, then it actually converges uniformly in every half-plane {s >. + e}, e > 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 ofBeurling primes for which bothBohr'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.