A note on Bohr's theorem for Beurling integer systems
type: Item
instance of: scholarly article
title: A note on Bohr's theorem for Beurling integer systems
author: Frederik Broucke
author: Q6052139
author: Karl-Mikael Perfekt
published in: Mathematische Annalen
publication date: 2024-08-06T00:00:00Z
review text: 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)].
reviewed by: Sami Omar
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11N80
Mathematics Subject Classification ID: 30B50
zbMATH DE Number: 7892694
zbMATH Keywords: Beurling numbers
zbMATH Keywords: Bohr's condition
MaRDI profile type: Publication
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DOI: 10.1007/S00208-023-02756-X
description: scientific article; zbMATH DE number 7892694