On a function related to the primes counting function

We study the function $\A(x)=\log x-\frac{x}{\pi(x)}$, where $\pi(x)$ is the number of primes not exceeding $x$. We find unconditional and also conditional bounds for it under assumption of the Riemann hypothesis, and we compute exact values of $\max\A(x)$ and $\inf\A(x)$ for $x\geqslant 2$.