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)][2] proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while [Pintz (1980)][1] 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.


  [1]: https://bibliotekanauki.pl/articles/1393142
  [2]: https://bibliotekanauki.pl/articles/1395100