Iterative maps emerging from cohomological structure of primes

TL;DR

This paper uncovers a cohomological structure underlying prime numbers, modeling prime gaps with an iterative map and linking prime irregularities to physical systems.

Contribution

It introduces a cohomological framework that describes prime gaps and fluctuations, connecting prime distribution to deterministic structures and the logarithmic integral function.

Findings

Prime gaps follow a function depending on separation distance.

A cohomological structure explains prime fluctuations.

The logarithmic integral function solves the cohomological equation.

Abstract

Prime numbers appeared in contexts spanning statistical mechanics, quantum mechanics and dynamical systems. However, the mechanisms governing the irregularities observed in their sequence and linking them to physical systems remained unclear. Here, it is shown that prime gaps at different separation distances follow a function depending on that distance and can be described by an iterative map which predicts the primary growth of successive primes. On the other hand, the analysis of remaining fluctuations reveals the existence of a well-defined cohomological structure, where the deterministic functional relation holds for primes up to small decaying fluctuations. In consequence, the long-range correlations as well as local jumps in primes encode the underlying cohomological structure where prime numbers are states of a given system that becomes deterministic asymptotically. Remarkably, the solution to this cohomological equation turns out to be the logarithmic integral function.