Independence in Mathematics -- the key to a Gaussian law

TL;DR

This paper explores the concept of statistical independence in mathematics, especially its role in number theory and the Gaussian law of errors, highlighting historical context, foundational assumptions, and modern developments.

Contribution

It revisits the notion of independence beyond Kolmogorov axioms, illustrating its significance in number theory and presenting recent advances in lacunary series related to independence.

Findings

Independence of binary expansion coefficients

Independence of divisibility by primes

Connection to Erdős-Kac central limit theorem

Abstract

In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the $1930$s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. We present the independence of the coefficients in a binary expansion, the independence of divisibility by primes, and the resulting, famous central limit theorem of Paul Erd\H{o}s and Mark Kac on the number of different prime factors of a number $n\in\mathbb N$. We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Rapha\"el Salem and Antoni Zygmund.