Bulletin of the Belgian Mathematical Society - Simon Stevin

Chebyshev Upper Estimates for Beurling's Generalized Prime Numbers

Jasson Vindas

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Abstract

Let $N$ be the counting function of a Beurling generalized number system and let $\pi$ be the counting function of its primes. We show that the $L^{1}$-condition $$ \int_{1}^{\infty}\left|\frac{N(x)-ax}{x}\right|\frac{\mathrm{d}x}{x}<\infty $$ and the asymptotic behavior $$N(x)=ax+O\left(\frac{x}{\log x}\right)\: ,$$ for some $a>0$, suffice for a Chebyshev upper estimate $$ \frac{\pi(x)\log x}{x}\leq B<\infty\: .$$

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1 (2013), 175-180.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1366306723

Digital Object Identifier
doi:10.36045/bbms/1366306723

Mathematical Reviews number (MathSciNet)
MR3082752

Zentralblatt MATH identifier
1280.11059

Subjects
Primary: 11N80: Generalized primes and integers
Secondary: 11N05: Distribution of primes 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Keywords
Chebyshev upper estimates Beurling generalized primes

Citation

Vindas, Jasson. Chebyshev Upper Estimates for Beurling's Generalized Prime Numbers. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 1, 175--180. doi:10.36045/bbms/1366306723. https://projecteuclid.org/euclid.bbms/1366306723


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