Following a previous question (here you'll find an introduction):
The book states that almost surely $$\pi_S(x+y)-\pi_S(x)=\mathrm{li}(x+y)-\mathrm{li}(x)+O(\sqrt y)$$ as soon as $y/(\log x)^2\to\infty$, with $y\le x$.
I couldn't prove this.
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Assuming the Riemann hypothesis – reuns Jun 14 '17 at 13:56