Multiple integral formulas for weighted zeta moments: the case of the sixth moment

We prove exact formulas for weighted 2kth moments of the Riemann zeta function for all integer k >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\geqslant 1$$\end{document} 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).