Write $\psi(x) = \sum_{n\le x} \Lambda(n)$. The classical omega theorem says that

$\psi(x) - x = \Omega_{\pm}(x^{1/2})$.

Question: How often does this hold? For example, what do we know about the size of the set

$\{ n\le x: \psi(x) - x > c x^{1/2} \}$

for some $c$? Ditto for $\psi(x) - x < c x^{1/2}$. What about replacing e.g. $x^{1/2}$ by $x^{\alpha}$ for some fixed $0 < \alpha < 1/2$? What's a good reference for such results? Thanks!