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

2023/11/05 by Sébastien Darses, Darses, Sébastien, Joseph Najnudel +1 · 1 citation
Mathematics · #11M06 #11M50 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2311.02783

openalex publication_date 2023/11/05 · openalex created_date 2023/11/08 · openalex updated_date 2026/07/28

Abstract

We prove exact formulas for weighted 2kth moments of the Riemann zeta function for all integer k≥ 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 Γ^kζ^k, which enjoy remarkable properties that are not satisfied by the more classically studied structure Γζ^k.

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