Machine learning of the prime distribution

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 Y.–H. 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.