Conditions for Beurling's integers to have a density

TL;DR

This paper investigates conditions under which Beurling's generalized integers have a density, providing new proofs of existing conditions, examples showing their near necessity, and employing Fourier analysis and probability methods.

Contribution

It offers a new proof of Diamond's condition for density and demonstrates that this condition is nearly necessary, expanding understanding of Beurling's generalized integers.

Findings

New proof of Diamond's condition for density

Examples showing the condition is not strictly necessary

The condition is very close to being necessary

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.