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7. Pair correlation


    1. Add remark

      Problem 7.1.

      [M. Radziwill] We can get lower bounds for the Type I off-diagonal terms using sifted sequences. Is there any analog of this for Type II terms? Can one improve lower bounds for $F(\alpha,T)$ in this way (as in Goldston-Gonek-Ozluk-Snyder)?
        1. Remark. [K. Soundararajan] Can one get lower bounds for moments in general?


            • Add remark

              Problem 7.2.

              [J. Keating] Compute the $3$-point correlation function $R_3(x_1,x_2,x_3)$ for zeta rigorously under suitable hypotheses (e.g. Hardy-Littlewood conjecture for twin primes).


                • Add remark

                  Problem 7.3.

                  [J. Keating] Compute the $4$-point correlation function rigorously under suitable hypotheses (here, Type II terms come in).

                      Cite this as: AimPL: Moments of zeta and correlations of divisor sums, available at http://aimpl.org/zetamoments.