Twin prime correlations from the pair correlation of Riemann zeros

J. P. Keating*, D. J. Smith

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
81 Downloads (Pure)


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 E → ∞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.
Original languageEnglish
Article number365201
Number of pages12
JournalJournal of Physics A: Mathematical and Theoretical
Issue number36
Early online date13 Aug 2019
Publication statusPublished - 13 Aug 2019


  • number theory
  • quantum chaos
  • random matrix theory


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