On coefficients satisfying Chebyshev's approximation of $\pi(x)$

Abstract

We note an interesting and under-expressed fact from Chebyshev's initial bounding for the prime counting function, $\pi(x) := \# \{p \leq x : p \text{ prime}\},$ based upon a selection of fixed coefficients $d\in D$ to show $\psi(x) \asymp x$, and thus the goal of choosing some $a(d)$ approximately the same as $\mu(d)$ such that: $$ \sum_{d}\frac{a(d)}{d} = 0, \quad \wedge \quad -\sum_{d}\frac{a(d)\log d}{d} \approx 1.$$