Expected biases in the distribution of consecutive primes

TL;DR

This paper explains observed biases in prime gaps modulo 10 as a natural consequence of the evolution of gaps during Eratosthenes sieve, showing these biases are transient and fade over time.

Contribution

It connects empirical biases in prime distributions to the theoretical evolution of gaps in Eratosthenes sieve, providing a unified explanation.

Findings

Biases in prime gaps are due to early-stage dynamics of the sieve.

Observed biases diminish and become uniform as the sieve progresses.

Theoretical results align with computational observations beyond current ranges.

Abstract

In 2016 Lemke Oliver and Soundararajan examined the gaps between the first hundred million primes and observed biases in their distributions modulo 10. Given our work on the evolution of the populations of various gaps across stages of Eratosthenes sieve, the observed biases are totally expected. The biases observed by Lemke Oliver and Soundararajan are a wonderful example for contrasting the computational range with the asymptotic range for the populations of the gaps between primes. The observed biases are the combination of two phenomena: (a) very small gaps, say $2 \le g \le 30$, get off to quick starts and over the first 100 million primes larger gaps are too early in their evolution; and (b) the assignment of small gaps across the residue classes disadvantages some of those classes - until enormous primes, far beyond the computational range. For modulus 10 and a few other bases, we aggregate the gaps by residue class and track the evolution of these teams as Eratosthenes sieve continues. The relative populations across these teams start with biases across the residue classes. These initial biases fade as the sieve continues. The OS enumeration strongly agrees with a uniform sampling at the corresponding stage of the sieve. The biases persist well beyond the computational range, but they are ultimately transient.