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, $\pi (x):=\#\{p\le x:p\text{ prime}\},$ 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: $$\sum _d\frac{a(d)}{d}=0,\wedge -\sum _d\frac{a(d)\log d}{d}\approx 1.$$

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