The $S$-$E$ route to the Chebyshev bounds for the prime-counting function

Abstract

We introduce the weighted prime sum $S(x) = \sum_{p \le x} \sqrt{(\log p)/p}$ and the derived quantity $E(x) = S(x)^2 - M(x)$, where $M(x) = \sum_{p \le x} (\log p)/p$. We prove that the order-of-magnitude estimate $S(x) \asymp \sqrt{x / \log x}$ implies the Chebyshev bounds $\pi(x) \asymp x / \log x$ through a short and transparent chain of inequalities. The mechanism passes through $E(x)$, which we show satisfies $E(x) \asymp \pi(x)$ whenever the size estimate for $S(x)$ holds. We also establish that $S(x) \asymp \sqrt{x / \log x}$ follows from the classical estimate $\sum_{p \le x} (\log p)/p = \log x + O(1)$ (Mertens' theorem), so the entire argument is self-contained. The result itself (the Chebyshev bounds) is classical, but the proof route through the $S$-$E$ mechanism appears to be new.