On Diamond's $L^1$ criterion for asymptotic density of Beurling generalized integers

Abstract

We give a short proof of the $L^1$ criterion for Beurling generalized integers to have a positive asymptotic density. We actually prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate $m(x)={\sum }_{n_k\le x}\mu (n_k)/n_k=o(1)$, with $\mu$ the Beurling analog of the Moebius function.

13 pages

References in corpus (2)

• On PNT equivalences for Beurling numbers
• Optimal Tauberian constant in Ingham's theorem for Laplace transforms