On the pair correlation conjecture and the alternative hypothesis

Abstract

We prove the equivalence of certain asymptotic formulas for (a) averages over intervals for the 2-point form factor F(α,T) for the zeros of the Riemann zeta-function, ζ(s), (b) the mean square of the logarithmic derivative of ζ(s), (c) a variance for the number of primes in short intervals, and (d) the number of pairs of zeros of ζ(s) with small gaps. The main result is a generalization of the fusion of a theorem of Goldston and a theorem of Goldston, Gonek, and Montgomery. We apply our result to deduce several consequences of the Alternative Hypothesis.