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Twin prime correlations from the pair correlation of Riemann zeros

Research output: Contribution to journalArticle

  • J. P. Keating
  • D. J. Smith
Original languageEnglish
Article number365201
Number of pages12
JournalJournal of Physics A: Mathematical and Theoretical
Issue number36
Early online date13 Aug 2019
DateAccepted/In press - 22 Jul 2019
DateE-pub ahead of print - 13 Aug 2019
DatePublished (current) - 13 Aug 2019


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.

    Research areas

  • number theory, quantum chaos, random matrix theory



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