Conditions for Beurling's integers to have a density

2016/11/14 by Jean‐Pierre Kahane, Kahane, Jean-Pierre
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Mathematics #Number Theory (math.NT) #Quantum Mechanics and Applications #Relativity and Gravitational Theory

paper · pdf · doi:10.48550/arxiv.1611.04432

openalex publication_date 2016/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1997 H.G.Diamond gave a condition on Beurling's generalized prime numbers in order that the corresponding generalized integers have a density. We give a new proof of this condition (Theorem 1) and a proof that it is not necessary (Theorem 2 and Examples). However, it is very near to be necessary (Theorem 3). Both proofs of Theorems 1 and 2 rely on Fourier analysis, mainly the Wiener algebra, and partly on probability methods.

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