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

Abstract

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 in Number Theory, we obtain a heuristic expression applicable when h N but h N, providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.

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