arXiv subjects
Publications and source records attributed to Hugh L. Montgomery.
We show assuming RH that phenomena concerning pairs of zeros established $via$ pair correlations occur with positive density (with at most a slight adjustment of the constants). Also, while a double zero is commonly considered to be a close pair, we consider the difference between two $distinct$ zeros.
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 $ψ(x+H)- ψ(x)$, for $0\le x\le N$, is approximately normal with mean $\sim H$ and variance $\sim H\log N/H$, when $N^δ\le H \le N^{1-δ}$.
The authors study the distribution of psi(x+h)-psi(x)-h and compare it with numerical data.