# The primes are not metric Poissonian

## TL;DR

This paper demonstrates that the distribution of fractional parts of primes does not satisfy the stronger pair correlation property, challenging assumptions about their second-order equidistribution.

## Contribution

It proves that primes lack the second-order equidistribution property known as metric Poissonian, contrasting with the classical first-order equidistribution results.

## Key findings

- Primes are not metric Poissonian.
- Second-order equidistribution does not hold for primes.
- Challenges assumptions in additive combinatorics related to primes.

## Abstract

It has been known since Vinogradov that, for irrational $\alpha$, the sequence of fractional parts $\{\alpha p\}$ is equidistributed in $\mathbb{R}/\mathbb{Z}$ as $p$ ranges over primes. There is a natural second-order equidistribution property, a pair correlation of such fractional parts, which has recently received renewed interest, in particular regarding its relation to additive combinatorics. In this paper we show that the primes do not enjoy this stronger equidistribution property.