We prove exact formulas for weighted 2kth moments of the Riemann zeta function for all integer \(k\geqslant 1\) in terms of the analytic continuation of an auto-correlation function. This latter enjoys three functional equations. One of them, following from a fundamental lemma of Bettin and Conrey (Algebra Number Theory 7(1):215–242, 2013), yields to a new formula for the sixth moment, which can be seen as a generalization of formulas by Titchmarsh (Proc Lond Math Soc 27(2):137–150, 1927) for the second and fourth moments. A basic and powerful tool is a special Fourier transform unveiled by Ramanujan (Quart J Math 46:253–260, 1915).
