Chebyshev Estimates for Beurling Generalized Prime Numbers. I

Abstract

We provide new sufficient conditions for Chebyshev estimates for Beurling generalized primes. It is shown that if the counting function $N$ of a generalized number system satisfies the $L^{1}$-condition $$ \int_{1}^{\infty}|\frac{N(x)-ax}{x}|\frac{\mathrm{d}x}{x}<\infty $$ and $N(x)=ax+o(x/\log x),$ for some $a>0$, then $$ 0<\liminf_{x\to\infty}\frac{\psi(x)}{x}\ \ \ {and}\ \ \ \limsup_{x\to\infty}\frac{\psi(x)}{x}<\infty $$ hold. We give an analytic proof of this result. It is based on Wiener division theorem. Our result extends those of Diamond (Proc. Amer. Math. Soc. 39 (1973), 503--508) and Zhang (Proc. Amer. Math. Soc. 101 (1987), 205--212).