A note on Bohr’s theorem for Beurling integer systems

Abstract

Given a sequence of frequencies $$\{\lambda _n\}_{n\ge 1}$$ { λ n } n ≥ 1 , a corresponding generalized Dirichlet series is of the form $$f(s)=\sum _{n\ge 1}a_ne^{-\lambda _ns}$$ f ( s ) = ∑ n ≥ 1 a n e - λ n s . We are interested in multiplicatively generated systems, where each number $$e^{\lambda _n}$$ e λ n arises as a finite product of some given numbers $$\{q_n\}_{n\ge 1}$$ { q n } n ≥ 1 , $$1 < q_n \rightarrow \infty $$ 1 < q n → ∞ , referred to as Beurling primes. In the classical case, where $$\lambda _n = \log n$$ λ n = log n , Bohr’s theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane $$\{\Re s> \theta \}$$ { ℜ s > θ } , then it actually converges uniformly in every half-plane $$\{\Re s> \theta +\varepsilon \}$$ { ℜ s > θ + ε } , $$\varepsilon >0$$ 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.