The identity contained in the last two displayed equations in the following passage (from page 110 in Ayoub's An Introduction to the Analytic Theory of Numbers, 1963) gives us right away a simple proof that the prime number theorem implies $M(x) = o(x)$. (Here $M(x)$ is the Mertens function $\sum...

Let $\mu$ be the Möbius function. Say we have a bound on $\check{M}(x) = \sum_{n\leq x} \mu(n) \log \frac{x}{n}$ of the form $|\check{M}(x)|\leq \epsilon x$ for all $x\geq x_0$. It is then easy to prove a bound $|M(x)|\leq 2\sqrt{\epsilon} x$ (plus a tiny error term) by a very simple procedure th...