Return to Revisions

For every $\varepsilon>0$, there exists $X_0(\varepsilon)>0$ such that for any $X>X_0(\varepsilon)$ we have $$\sup_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>0>\inf_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}.$$ 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.