  Recently, Michel Talagrand computed the large deviations limit
$\lim_{N\to\infty}(Na)^{-1}\log \e Z_N^a$ for the moments of the partition
function $Z_N$ in the Sherrington-Kirkpatrick model for all real $a\geq 0.$ For
$a\geq 1$ the limit is given by Guerra's inverse bound and this result extends
the classical physicist's replica method that corresponds to integer $a.$ We
give a new proof for $a\geq 1$ in the case of the pure $p$-spin SK model that
provides a strong exponential control of the overlap.
