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On Diamond’s L¹ criterion for asymptotic density of Beurling generalized integers

Gregory Debruyne (UGent) and Jasson Vindas Diaz (UGent)
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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 in fact prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate m(x)=∑_{n_k≤x} μ(n_k)/n_k=o(1) with the Beurling analog μ of the Möbius function.
Keywords
Asymptotic density, Beurling generalized numbers, Beurling primes, Moebius function, zeta function

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Please use this url to cite or link to this publication:

Chicago
Debruyne, Gregory, and Jasson Vindas Diaz. 2019. “On Diamond’s L1 Criterion for Asymptotic Density of Beurling Generalized Integers.” Michigan Mathematical Journal.
APA
Debruyne, G., & Vindas Diaz, J. (2019). On Diamond’s L1 criterion for asymptotic density of Beurling generalized integers. MICHIGAN MATHEMATICAL JOURNAL.
Vancouver
1.
Debruyne G, Vindas Diaz J. On Diamond’s L1 criterion for asymptotic density of Beurling generalized integers. MICHIGAN MATHEMATICAL JOURNAL. 2019;
MLA
Debruyne, Gregory, and Jasson Vindas Diaz. “On Diamond’s L1 Criterion for Asymptotic Density of Beurling Generalized Integers.” MICHIGAN MATHEMATICAL JOURNAL (2019): n. pag. Print.
@article{8606707,
  abstract     = {We give a short proof of the L\^{ }1 criterion for Beurling generalized integers to have a positive asymptotic density. We in fact prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate m(x)=\ensuremath{\sum}\_\{n\_k\ensuremath{\leq}x\} \ensuremath{\mu}(n\_k)/n\_k=o(1) with the Beurling analog \ensuremath{\mu} of the M{\"o}bius function.},
  author       = {Debruyne, Gregory and Vindas Diaz, Jasson},
  issn         = {0026-2285},
  journal      = {MICHIGAN MATHEMATICAL JOURNAL},
  language     = {eng},
  title        = {On Diamond{\textquoteright}s L{\textonesuperior} criterion for asymptotic density of Beurling generalized integers},
  url          = {http://dx.doi.org/10.1307/mmj/1548903624},
  year         = {2019},
}

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