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

Question

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?

Answer 1

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

Answer 2

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.

Answer 3

There are several 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). An English translation is here.