M(x)=o(x) Estimates for Beurling numbers
Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 469-483.

In classical prime number theory there are several asymptotic formulas said to be “equivalent” to the PNT. One is the bound M(x)=o(x) for the sum function of the Moebius function. For Beurling generalized numbers, this estimate is not an unconditional consequence of the PNT. Here we give two conditions that yield the Beurling version of the M(x) bound, and examples illustrating failures when these conditions are not satisfied.

Dans la théorie des nombres premiers classique, certaines expressions sont considérées comme « équivalentes » au TNP (Théorème des nombres premiers). Parmi elles on trouve la borne M(x)=o(x) pour la fonction sommatoire de Moebius. Dans le cas des nombres premiers, généralisés par Beurling, cette borne ne suit pas nécessairement du TNP sans exiger des hypothèses additionelles. Ici, deux conditions sont présentées, impliquant la version Beurling pour la borne sur M(x) et quelques exemples sont construits, démontrant l’absence éventuelle de cette borne si ces conditions ne sont pas réalisées.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1034
Classification: 11N80,  11M41
Keywords: Beurling generalized numbers; mean-value vanishing of the Moebius function; Chebyshev bounds; zeta function; prime number theorem; PNT equivalences
Gregory Debruyne 1; Harold G. Diamond 2; Jasson Vindas 1

1 Department of Mathematics Ghent University Krijgslaan 281 B 9000 Ghent, Belgium
2 Department of Mathematics University of Illinois 1409 W. Green St. Urbana IL 61801, U.S.A.
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Gregory Debruyne; Harold G. Diamond; Jasson Vindas. $M(x)=o(x)$ Estimates for Beurling numbers. Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 469-483. doi : 10.5802/jtnb.1034. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1034/

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