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A numerical study on the distributions of primes in short intervals of length$h$ over the natural numbers $N$ is presented. Based on Cram\'er's model inNumber Theory, we obtain a heuristic expression applicable when $h \gg \log{N}$but $h \ll N$, providing support to the Montgomery and Soundararajan conjectureon the variance of the prime distribution at this scale.

A heuristic study of the distribution of primes in short and not-so-short intervals