The probability that two random integers are coprime

TL;DR

This paper establishes a surprising equivalence between the Riemann Hypothesis and the rate at which the probability that two geometric random integers are coprime converges to 6/π² as the distribution parameter approaches zero.

Contribution

It demonstrates a novel connection between a major unsolved problem in mathematics and the asymptotic behavior of coprimality probabilities for geometric random variables.

Findings

Proves the equivalence between the Riemann Hypothesis and convergence speed of coprimality probability

Shows that the probability approaches 1/ζ(2) as the parameter tends to zero

Provides a new probabilistic perspective on the Riemann Hypothesis

Abstract

An equivalence is proven between the Riemann Hypothesis and the speed of convergence to 1/zeta(2) of the probability that two independent random variables following the same geometric distribution are coprime integers, when the parameter of the distribution goes to 0.