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

We note an interesting and under-expressed fact from Chebyshev’s initial bounding for the prime counting function, $\pi(x) := # {p \leq x : p \text{ prime}},$baseduponaselectionoffixedcoefficients$d\in D$toshow$\psi(x) \asymp x$,andthusthegoalofchoosingsome$a(d)$ approximately the same as $\mu (d)$ such that: $$\sum _d\frac{a(d)}{d}=0,\wedge -\sum _d\frac{a(d)\log d}{d}\approx 1.$$