In Wikipedia article about Mertens function we can read: "A curious relation given by Mertens himself involving the second Chebyshev function is
$$\displaystyle \psi (x)=\sum _{n=1}^{\infty } M\left({\frac {x}{n}}\right)\log n=M\left({\frac {x}{2}}\right)\log 2+M\left({\frac {x}{3}}\right)\log 3+M\left({\frac {x}{4}}\right)\log 4+\cdots."$$
How to prove it?
$$\Lambda (n)={\begin{cases}\log p&{\text{if }}n=p^{k}{\text{ for some prime }}p{\text{ and integer }}k\geq 1\\0&{\text{otherwise}}\end{cases}}$$
$$\mu(n) ={\begin{cases} +1& {\text{if n is a square-free positive integer with an even number of prime factors}} \\−1& {\text{if n is a square-free positive integer with an odd number of prime factors}}\\0& {\text{if n has a squared prime factor}}\end{cases}}$$
$$\displaystyle \psi (x)=\sum _{n=1}^x \Lambda (n)$$
$$\displaystyle M(x)=\sum _{n=1}^{x}\mu (n)$$