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 $${\int }_1^{\infty }|\frac{N(x)-ax}{x}|\frac{\mathrm{d}x}{x}<\infty$$ and the asymptotic behavior $$N(x)=ax+O(\frac{x}{\log x}),$$ for some $a>0$, suffice for a Chebyshev upper estimate $$\frac{\pi (x)\log x}{x}\le B<\infty .$$

5 pages