  We prove exact formulas for weighted $2k$th moments of the Riemann zeta
function for all integer $k\geq 1$ in terms of the analytic continuation of an
auto-correlation function. This latter enjoys several functional equations. One
of them, following from a fundamental lemma of Bettin and Conrey (2013), yields
to new formulas for the moments: our second main result is the case $k=3$, but
there is no obstruction to obtain higher moments. This generalizes results by
Titchmarsh (1928) for $k=1$ and $k=2$. A basic and powerful tool is a special
Fourier transform unveiled by Ramanujan (1915). In a nutshell, the new idea is
to consider the associated structures to $\Gamma^k\zeta^k$, which enjoy
remarkable properties that are not satisfied by the more classically studied
structure $\Gamma\zeta^k$.
