Pseudorandomness of primes at large scales

TL;DR

This paper investigates the distribution of primes at large scales, assuming a variant of the prime k-tuple conjecture, and shows that primes behave like a Poisson process under these assumptions.

Contribution

It introduces a method to compute mixed moments of primes in short intervals and progressions, advancing understanding of prime pseudorandomness at large scales.

Findings

Primes in disjoint intervals follow a Poisson process under the conjecture.

Established convergence of normalized prime counts to a Poisson distribution.

Estimated mean of singular series along lattice products.

Abstract

Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along products of lattices, which is of independent interest. As a consequence, we establish the convergence of both sequences of suitably normalized primes to a standard Poisson point process.