A question about an asymptotic formula

Question

I've been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann's hypothesis is true, but I was unable to find a journal reference for this. Does anybody know of any journal reference or any other source where I can find this conditional result?

Answer 1

I don't know the original source, but you can find the result in

• Helmut Maier, Primes in short intervals, Michigan Math. J. 32:2 (1985), pp. 131–255.

There are more details in

• D. R. Heath-Brown, The number of primes in a short interval, Journal für die reine und angewandte Mathematik (Crelles Journal), 389 (1988), pp. 22–63.

which proves a weaker result without RH.

Selberg had earlier proved that this, and more, is true (unconditionally) for a set of density 1:

• A. Selberg, An elementary proof of the Prime Number Theorem, Ann. of Math. 50 (1949), pp. 305–313.