Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution

TL;DR

This paper studies the average behavior of Euler products related to prime number conjectures, showing their limiting distributions and suggesting that prime patterns in short intervals follow a Poisson distribution.

Contribution

It introduces methods to compute averages of Euler products, demonstrates the distribution of singular series for prime tuples, and links prime patterns to Poisson distribution under certain conditions.

Findings

Singular series for prime tuples have a limiting distribution.

Moments of the singular series exhibit a symmetry property.

Prime patterns in short intervals are conjectured to be Poisson distributed.

Abstract

We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the $k$-tuple conjecture and more general problems of polynomial representation of primes. We show that the singular series for the $k$-tuple conjecture have a limiting distribution when taken over $k$-tuples with (distinct) entries of growing size, and observe that its moments have a curious symmetry property. We also give conditional arguments that would imply that the number of twin primes (or more general polynomial prime patterns) in suitable short intervals are asymptotically Poisson distributed.