A note on Bohr's theorem for Beurling integer systems

Abstract

Given a sequence of frequencies $\{{\lambda }_n{\}}_{n\ge 1}$, a corresponding generalized Dirichlet series is of the form $f(s)={\sum }_{n\ge 1}{a}_n{e}^{-{\lambda }_{n}s}$. We are interested in multiplicatively generated systems, where each number $e^{{\lambda }_n}$ arises as a finite product of some given numbers $\{q_n{\}}_{n\ge 1}$, $1<q_n\to \infty$, 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 $\{\Re s>\theta +\epsilon \}$, $\epsilon >0$. 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.

References in corpus (2)

• Beurling integers with RH and large oscillation
• Convergence and almost sure properties in Hardy spaces of Dirichlet series