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 $\pi(x) \asymp \bigl(\sum_{p \le x} \sqrt{\ln p / p}\bigr)^{2} \asymp x/\ln x$. 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 $\bigl(\sum_{p \le x} \sqrt{\ln p/p}\bigr)^{2} - \sum_{p \le x} (\ln p)/p$, which is shown as having the same order as $\pi(x)$.