Primes in short intervals

Abstract

Contrary to what would be predicted on the basis of Cramér's model concerning the distribution of prime numbers, we develop evidence that the distribution of $\psi (x+H)-\psi (x)$, for $0\le x\le N$, is approximately normal with mean $\sim H$ and variance $\sim H\log N/H$, when $N^{\delta }\le H\le N^{1-\delta }$.

29 pages