Estimating constellations among primes - I. Uniformity

TL;DR

This paper develops a recursive method to estimate the frequency of specific prime constellations, such as consecutive primes in arithmetic progression, by assuming uniform distribution among generators in Eratosthenes sieve.

Contribution

It introduces a recursion-based approach to enumerate and estimate prime constellations, incorporating a uniformity assumption to predict their asymptotic behavior.

Findings

Estimates align with computational data for initial cases.

Systematic error correlates with constellation length.

Method shows correct asymptotic trends.

Abstract

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations. In this paper, we use those enumerations to estimate the numbers of these constellations that occur as constellations among prime numbers, and we compare these estimates with computational results. We include in our estimates the constellations corresponding to three and four consecutive primes in arithmetic progression. For these initial estimates, we assume that the copies of a given constellation tend toward a uniform distribution in the cycle of gaps, as the recursion progresses. Our simple estimates based on the recursion of gaps and the assumption of uniformity appear to have correct asymptotic behavior, and they exhibit a systematic error correlated to length of the constellation.