Jumping champions and prime gaps using information-theoretic tools

TL;DR

This paper applies information-theoretic Shannon sampling methods to analyze prime gaps, revealing that Fourier transforms of constructed signals spike at prime spacings, especially at primorials, indicating their role as jumping champions.

Contribution

It introduces a novel application of Shannon sampling theory to prime number analysis, highlighting early signals of primorials as jumping champions in prime gaps.

Findings

Fourier transforms spike at prime spacings

Primorials show prominent early spikes

Supports primorials as jumping champions

Abstract

We study the spacing of the primes using methods from information theory. In information theory, the equivalence of continuous and discrete representations of information is established by Shannon sampling theory. Here, we use Shannon sampling methods to construct continuous functions whose varying bandwidth follows the distribution of the prime numbers. The Fourier transforms of these signals spike at frequently occurring spacings between the primes. We find prominent spikes, in particular, at the primorials. Previously, the primorials have been conjectured to be the most frequent gaps between subsequent primes, the so-called "jumping champions". Here, we find a foreshadowing of the primorial's role as jumping champions in the sense that Fourier spikes for the primorials arise much earlier on the number axis than where the primorials in question are expected to reign as jumping champions.