An Equivalent Form of the Prime Number Theorem - Reddy, G. Sudhaamsh Mohan; Rau, S. Srinivas; Uma, B. (Sarajevo Journal of Mathematics, 2020)
A simple proof is given that $\sum\limits_{n}\frac{μ(n) d(n)}{n}=0$ using the Prime Number Theorem. It is shown that this is equivalent to the estimate $\sum\limits_{n≤ x}μ(n)d(n)=°(x)$ and to the Prime Number Theorem.