A note on Bohr's theorem for Beurling integer systems
The Bohr condition for frequencies \(\lbrace \lambda_n \rbrace_{n\geq 1}\) asks for the existence of \(c_1, c_2 > 0\) such that \N\[\N\lambda_{n+1} - \lambda_n \geq c_1e^{-c_2\lambda_{n+1}},\quad n\in \mathbb{N}.\N\]\NLet \(\lbrace p_n\rbrace_{n\geq 1}\) be the sequence of ordinary prime numbers and let \(N \geq 1\). Then, it is shown that Bohr's condition holds for the Beurling integers generated by the Beurling primes \(q=\lbrace p_n\rbrace_{n\geq 1}\cup \lbrace q_j\rbrace_{j=1}^N\) for almost every choice \((q_1,\ldots, q_N ) \in (1,\infty )^N\). Applying this result in conjunction with a probabilistic method, the authors find a system of Beurling primes for which both Bohr's condition and the Riemann hypothesis are valid. This yields a counterexample to a conjecture of \textit{H. Helson} concerning outer functions in Hardy spaces of generalized Dirichlet series [Ark. Mat. 8, 139--143 (1970; Zbl 0199.46601)].
scientific article; zbMATH DE number 7892694
scientific article