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

Gregory Debruyne; Jasson Vindas

doi:10.1307/mmj/1548903624

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

Number Theory 2019-08-13 v2 Classical Analysis and ODEs

Authors: Gregory Debruyne , Jasson Vindas

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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 $m(x)=\sum_{n_{k}\leq x} \mu(n_k)/n_k=o(1)$ , with $\mu$ the Beurling analog of the Moebius function.

Keywords

Beurling generalized integers asymptotic density L^1 criterion Beurling Moebius function density existence sufficient conditions

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@article{arxiv.1704.03771,
  title  = {On Diamond's $L^1$ criterion for asymptotic density of Beurling generalized integers},
  author = {Gregory Debruyne and Jasson Vindas},
  journal= {arXiv preprint arXiv:1704.03771},
  year   = {2019}
}

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13 pages

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

Abstract

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