Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if we apply Perron's formula directly, without going through $\psi(x)$, we end up with an expression for $\pi(x)$ in terms of an integral involving $\log \zeta(s)$. Here we see what the issue is: $\log \zeta(s)$ has a branch point at $s=1$, and that is unpleasant.

At the same time, it is not *that* unpleasant; you can do a little loop ("truncated Hankel contour") around $s=1$. Is there a standard reference where $\pi(x)$ is estimated in this way?