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!



How often does the omega theorem hold?
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)](https://bibliotekanauki.pl/articles/1395100) proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while [Pintz (1980)](https://bibliotekanauki.pl/articles/1393142) 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.


