TITLE: How often does the omega theorem hold?
QUESTION [6 upvotes]: 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!

REPLY [7 votes]: For any $\varepsilon>0$, there exist $c(\varepsilon)>0$ and $X_0(\varepsilon)>0$ such that for any $X>X_0(\varepsilon)$ we have
$$\sup_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>c(\varepsilon)\qquad\text{and}\qquad
\inf_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}<-c(\varepsilon).$$
More precisely, Ingham (1935) proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while Pintz (1980) proved a stronger result for the case when the real parts of the zeta zeros do not have a maximum. There might be even stronger results in the literature, please check.