7. Pair correlation

Problem 7.1.
[M. Radziwill] We can get lower bounds for the Type I offdiagonal 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 GoldstonGonekOzlukSnyder)?
Remark. [K. Soundararajan] Can one get lower bounds for moments in general?


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. HardyLittlewood conjecture for twin primes). 
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.