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!