Chebyshev Upper Estimates for Beurling's Generalized Prime Numbers

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 $$ \frac{\pi(x)\log x}{x}\leq B<\infty\: .$$