Chebyshev Upper Estimates for Beurling's Generalized Prime Numbers

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

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