Define $N_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$.
Landau's Theorem says $N_\Delta(T)$ is $\Omega(\log T)$ [1].
But perhaps that estimate is too crude. Is the main term of $N_\Delta(T)$ known? Or are only strict upper and lower bounds known?
What type of machinery is used to determine something of this nature?
[1] Emil Grosswald "Oscillation Theorems of Arithmetical Functions" Transactions of the American Mathematical Society 126 (1967) pp. 7.