Berry's conjecture on the variance of zeta zeros

Let $S(t)$ be the argument function of the Riemann zeta function, and let $\gamma_0$ be Euler's constant. For $\delta>0$, consider the mean square of the increment $S(t+2\pi\delta/\log T)-S(t)$. Berry's conjecture. As $T\to\infty$, the following asymptotic formulae hold:

(a) If $\delta=o(\log T)$, then

$$\int_0^T\left[S\left(t+\frac{2\pi\delta}{\log T}\right)-S(t)\right]^2\,\mathrm{d}t=\frac{T}{\pi^2}\left[\log(2\pi\delta)-\operatorname{Ci}(2\pi\delta)-2\pi\delta\operatorname{Si}(2\pi\delta)+\pi^2\delta-\cos(2\pi\delta)+1+\gamma_0\right]+o(T).$$

(b) If $\delta\gg\log T$, then

$$\int_0^T\left[S\left(t+\frac{2\pi\delta}{\log T}\right)-S(t)\right]^2\,\mathrm{d}t=\frac{T}{\pi^2}\left[\sum_{n\leq T}\frac{\Lambda^2(n)}{n\log^2 n}\left(1-\cos\left(\frac{2\pi\delta\log n}{\log T}\right)\right)+1\right]+o(T).$$

Here $\Lambda$ is the von Mangoldt function, $\operatorname{Si}(x)=\int_0^x\frac{\sin u}{u}\,\mathrm{d}u$, and $\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos u}{u}\,\mathrm{d}u$. The universal regime in part (a) was proved by Fujii assuming RH and Montgomery's strong pair correlation conjecture. Assuming RH and Chan's longer-range version of that conjecture, the paper verifies both parts in the range $\delta=o(\log^{4/3}T)$; part (b) may hold in a longer range.