Oscillations of $\theta(x)-x$, for the Chebyshev $\theta$ function

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 small-scale sign changes (they are provably averaged away).

I'm not aware of any published upper bounds for $$W^\theta(T)$$.

*J. Kaczorowski, "On sign-changes in the remainder-term of the prime-number 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.

• This answers a different question than the one the OP asked. – 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), 21-88 (1996). English translation is here.