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 $$0<\underset{x\to \infty }{\mathrm{liminf}}\frac{\psi (x)}{x}and\underset{x\to \infty }{\mathrm{limsup}}\frac{\psi (x)}{x}<\infty$$ 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).

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