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?




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[1] Emil Grosswald "Oscillation Theorems of Arithmetical Functions" *Transactions of the American Mathematical Society* **126** (1967) pp. 7.



Main term in the number of sign changes of $\psi(x) - x$
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](http://matwbn.icm.edu.pl/ksiazki/aa/aa44/aa4446.pdf)). For recent work related to this, see [Montgomery and Vorhauer](https://www.projecteuclid.org/download/pdf_1/euclid.facm/1229442626), 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](http://www.ams.org/journals/tran/2009-361-09/S0002-9947-09-04803-X/S0002-9947-09-04803-X.pdf). 


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  

$$ 
0< \liminf\_{T\to \infty} \frac{N\_{\Delta}(T)}{\sqrt{T}} < \limsup\_{T\to\infty} \frac{N\_{\Delta}(T)}{\sqrt{T}} < \infty. 
$$


