Chebyshev Upper Estimates for Beurling's Generalized Prime Numbers

TL;DR

This paper establishes that certain integrability and asymptotic conditions on Beurling's generalized number system imply a Chebyshev upper estimate for the distribution of its primes.

Contribution

It proves that specific $L^1$-conditions and asymptotic behavior of the counting function guarantee Chebyshev upper bounds for generalized primes.

Findings

$L^1$-condition ensures prime counting upper bounds

Asymptotic form of $N(x)$ implies Chebyshev estimate

Provides conditions for prime distribution in generalized systems

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 $$ \frac{\pi(x)\log x}{x}\leq B<\infty. $$