Twin prime correlations from the pair correlation of Riemann zeros

Abstract

We establish, via a 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. An averaged form of the Hardy–Littlewood conjecture is obtained by inverting the 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.