Main term in the number of sign changes of $\psi(x) - x$ 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.
 A: An asymptotic for the number of sign changes is not known, and indeed only a lower bound of $c \log T$ is known.  There have been small improvements in the constant $c$ that is allowed here (see Kaczorowski).   For recent work related to this, see Montgomery and Vorhauer, who show that the remainder term changes sign in any interval $[x,2.02x]$ once $x$ is large enough.  They also speculate that the right order for the number of sign changes up to $T$ is $\sqrt{T}$  (every once in a while $\psi(x)-x$ will be very small, and near that point there'll be many sign changes).  One more recent reference is a paper of Kaczorowski and Wiertelak. 
Also one shouldn't really expect an asymptotic formula for the number of sign changes -- if $\psi(x)-x$ is large and positive (on scale $\sqrt{x}$)  then it is likely to stay positive at $1.01x$ as well.  Montgomery and Vorhauer speculate that
$$ 
0< \liminf_{T\to \infty} \frac{N_{\Delta}(T)}{\sqrt{T}} < \limsup_{T\to\infty} \frac{N_{\Delta}(T)}{\sqrt{T}} < \infty. 
$$
