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arXiv · 1903.07057

Twin prime correlations from the pair correlation of Riemann zeros

J. P. Keating·D. J. Smith
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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.

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J. P. Keating, D. J. Smith. 2019-03-17. Twin prime correlations from the pair correlation of Riemann zeros. https://doi.org/10.1088/1751-8121/ab3521

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