Twin prime correlations from the pair correlation of Riemann zeros

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 $E\to \infty$ 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.

11 pages

References in corpus (2)

• Moments of zeta and correlations of divisor-sums: V
• Averages of ratios of the Riemann zeta-function and correlations of divisor sums