Machine Learning of the Prime Distribution

TL;DR

Using maximum entropy methods to derive several theorems in probabilistic number theory, including a version of the Hardy–Ramanujan Theorem and a theoretical argument explaining the experimental observations of Y.–H.

Abstract

In the present work we use maximum entropy methods to derive several theorems in probabilistic number theory, including a version of the Hardy-Ramanujan Theorem. We also provide a theoretical argument explaining the experimental observations of Yang-Hui He about the learnability of primes, and posit that the Erdős-Kac law would very unlikely be discovered by current machine learning techniques. Numerical experiments that we perform corroborate our theoretical findings.