An Elementary Proof Of The Chebyshev Bounds Via A Weighted Prime Sum

Abstract

The Chebyshev bounds for the prime-counting function, i.e., $\pi (x)\asymp x/\ln x$, is established in a new way. This new approach follows the outline $\pi (x)\asymp ({\sum }_{p\le x}\sqrt{\ln p/p}{)}^2\asymp x/\ln x$. Here, the second $\asymp$ is derived from the classical estimate by Mertens, i.e., ${\sum }_{p\le x}(\ln p)/p=\ln x+O(1)$; while the first $\asymp$ is proved by considering the difference $({\sum }_{p\le x}\sqrt{\ln p/p}{)}^2-{\sum }_{p\le x}(\ln p)/p$, which is shown as having the same order as $\pi (x)$.

6 pages. Minor revision: title change and updated references