Scale invariant correlations and the distribution of prime numbers

TL;DR

This paper investigates the scale-invariant negative correlations in prime number distributions, compares them to fractional Brownian motion, and explores implications for the Riemann hypothesis.

Contribution

It introduces a novel analysis of prime correlations showing scale invariance and proposes using this as a test for the Riemann hypothesis.

Findings

Negative correlations in primes show scale invariance

Prime series resemble fractional Brownian motion with nonstationary behavior

Potential link between scale invariance and the Riemann hypothesis

Abstract

Negative correlations in the distribution of prime numbers are found to display a scale invariance. This occurs in conjunction with a nonstationary behavior. We compare the prime number series to a type of fractional Brownian motion which incorporates both the scale invariance and the nonstationary behavior. Interesting discrepancies remain. The scale invariance also appears to imply the Riemann hypothesis and we study the use of the former as a test of the latter.