Chebyshev Estimates for Beurling Generalized Prime Numbers. I

TL;DR

This paper establishes new sufficient conditions under which Chebyshev estimates hold for Beurling generalized primes, extending previous results by employing an analytic approach based on Wiener division theorem.

Contribution

It introduces novel $L^{1}$-conditions on the counting function that guarantee Chebyshev estimates for generalized primes, expanding the theoretical framework in this area.

Findings

Provides conditions ensuring positive lim inf and finite lim sup of $rac{\psi(x)}{x}$

Extends previous results by Diamond and Zhang

Uses Wiener division theorem for analytic proof

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).