An information-theoretic upper bound on prime gaps

TL;DR

This paper uses information theory to establish an upper bound on prime gaps, clarifying the relationship between prime distribution and entropy, and highlighting limits on machine learning approaches in number theory.

Contribution

It introduces an information-theoretic framework to analyze prime gaps and addresses Cramér's conjecture, revealing fundamental epistemic limits in applying machine learning to prime distribution.

Findings

Provides an upper bound on prime gaps based on entropy considerations

Establishes a connection between prime density and source distribution

Highlights limitations of machine learning in prime analysis

Abstract

Within the setting of rare event modelling, the method of level sets allows us to define an equivalence relation over rare events with distinct rates of entropy production. This method allows us to clarify the relation between the empirical density of primes and their source distribution which then allows us to address Cram\'er's conjecture, an open problem in probabilistic number theory. As a natural consequence, this analysis places strong epistemic limits on the application of machine learning to analyse the distribution of primes.