  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.
