# Twin prime correlations from the pair correlation of Riemann zeros

## TL;DR

This paper demonstrates that the Hardy-Littlewood twin prime conjecture is equivalent to the asymptotic behavior of the two-point correlation function of Riemann zeros, establishing a deep connection between prime pairs and zero correlations.

## Contribution

It proves the equivalence between the twin prime conjecture and the two-point correlation function of Riemann zeros, including a precise asymptotic formula with lower order terms.

## Key findings

- Hardy-Littlewood conjecture is equivalent to Riemann zero correlations
- Derived an asymptotic formula for the two-point correlation function
- Connected prime pair distribution with Riemann zero statistics

## Abstract

We establish, via a formal/heuristic Fourier inversion calculation, that the Hardy-Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height $E$ on the critical line. Previously it was known that the Hardy-Littlewood conjecture implies the pair correlation formula, and we show that the reverse implication also holds. A smooth form of the Hardy-Littlewood conjecture is obtained by inverting the $E \rightarrow \infty$ limit of the two-point correlation function and the precise form of the conjecture is found by including asymptotically lower order terms in the two-point correlation function formula.