Is anything known about the relative "periodicity" of the oscillations of $\theta(x)x$, that is, how frequent, in general terms, are the sign changes? Here, $\theta(x)$ is the Chebyshev $\theta$. If something is known, could you provide the reference to this?
Let $W^\theta(T)$ denote the number of sign changes of $\theta(x)x$ on the interval $[0,T]$. Kaczorowski* proved that $W^\theta(T) \gg \log T$, which I believe is the best known lower bound (if not, it's not far off). Roughly speaking, the proof proceeds by averaging $\theta(x)x$ many times over intervals and showing that the averaged version has many sign changes.
The truth is probably that $W^\theta(T)$ is closer to $\sqrt T$ in reality. Typically $\theta(x)x$ has order of magnitude $\sqrt x$; for intervals on which it is $o(\sqrt x)$, it presumably (by analogy to random walks) has many sign changes in each such interval, before moving on to a long interval on which it is large again. The averaging method cannot detect these smallscale sign changes (they are provably averaged away).
I'm not aware of any published upper bounds for $W^\theta(T)$.
*J. Kaczorowski, "On signchanges in the remainderterm of the primenumber formula. II", Acta Arith. 45 (1985), no. 1, 65–74. MR791085
In their paper Chebyshev's bias, Rubinstein and Sarnak showed, under the RH and another natural hypothesis, that the quantity $$\pi(x)  \text{Li}(x)$$ is positive for a proportion $$\approx 0.00000026$$ of the time, if you change your time scale to a logarithmic one. Since the difference $$\theta(x)x$$ oscillates according to the same distribution of zeroes (the ones of the Riemann $\zeta$ function of course), I am tempted to say the proportion would be very close to this.

2$\begingroup$ This answers a different question than the one the OP asked. $\endgroup$ – Greg Martin Mar 29 at 17:55
There are sevtral results on this topic in the article S. B. Stechkin, A. Yu. Popov, “The asymptotic distribution of prime numbers on the average”, Uspekhi Mat. Nauk, 51:6 (312), 2188 (1996). English translation is here.