  In this note we give two small results concerning the correlations of the
Selberg sieve weights. We then use these estimates to derive a new
(conditional) lower bound on the variance of the primes in short intervals, and
also on the so-called `form factor' for the pair correlations of the zeros of
the Riemann zeta function. Our bounds ultimately rely on the estimates of
Bettin--Chandee for trilinear Kloosterman fractions.
