  We examine the calculation of the second and fourth moments and shifted
moments of the Riemann zeta-function on the critical line using long Dirichlet
polynomials and divisor correlations. Previously this approach has proved
unsuccessful in computing moments beyond the eighth, even heuristically. A
careful analysis of the second and fourth moments illustrates the nature of the
problem and enables us to identify the terms that are missed in the standard
application of these methods.
