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On the estimate $M(x)=o(x)$ for Beurling generalized numbers

Number Theory 2025-06-10 v2

Abstract

We show that the sum function of the M\"{o}bius function of a Beurling number system must satisfy the asymptotic bound M(x)=o(x)M(x)=o(x) if it satisfies the prime number theorem and its prime distribution function arises from a monotone perturbation of either the classical prime numbers or the logarithmic integral.

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Cite

@article{arxiv.2406.00736,
  title  = {On the estimate $M(x)=o(x)$ for Beurling generalized numbers},
  author = {Jasson Vindas},
  journal= {arXiv preprint arXiv:2406.00736},
  year   = {2025}
}

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9 pages