Chebyshev Estimates for Beurling Generalized Prime Numbers. I

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 $${\int }_1^{\mathrm{\infty }}\mathrm{\mid }\frac{N(x)-ax}{x}\mathrm{\mid }\frac{\mathrm{d}x}{x}<\mathrm{\infty }$$ and $N(x)=ax+o(x\mathrm{/}\log x),$ for some $a>0$, then $$0<\underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\psi (x)}{x}and\underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\psi (x)}{x}<\mathrm{\infty }$$ 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).