## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Chebyshev Upper Estimates for Beurling's Generalized Prime Numbers

Jasson Vindas

#### 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