Is there any link to the paper " J. E. Littlewood, Sur la distribution des nombres premiers"? I cannot find anywhere on googling.
What about the analogous to the theorem for small interval? Is there anyway to say that the functions, for $x<x_1$, $\text{li}'(x,x_1):=\text{li}(x_1)-\text{li}(x)$ and $\pi'(x,x_1):=\pi(x_1)-\pi(x)$ possess the same property, which is, for instance, $\text{li}'(x/2,x)-\pi'(x/2,x)$ changes sign infinitely often?