Changes of sign of the error term in the prime number theorem

Abstract

We assume the Riemann Hypothesis (RH). It is classical that there is an absolute constant $C > 1$ such that $\psi(x)-x$ changes sign in every interval $[x, Cx]$ for $x\ge 1$. We prove that $\psi(x)-x$ changes sign in $[x,19x]$ for all $x\ge 1$, and also that for $x\ge x_0$, $\psi(x)-x$ changes sign in the interval $[x,Cx]$ where $C = 2.02$.