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

Sébastien Darses; Joseph  Najnudel

doi:10.1007/s11139-024-00944-0

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

Sébastien Darses^*, Joseph Najnudel

^*Corresponding author for this work

School of Mathematics

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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 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).

Original language	English
Article number	1421–1447
Number of pages	27
Journal	Ramanujan Journal
Volume	65
Issue number	3
Early online date	20 Sept 2024
DOIs	https://doi.org/10.1007/s11139-024-00944-0
Publication status	Published - 1 Nov 2024

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title = "Multiple integral formulas for weighted zeta moments: the case of the sixth moment",

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 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).",

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T1 - Multiple integral formulas for weighted zeta moments

T2 - the case of the sixth moment

AU - Darses, Sébastien

AU - Najnudel, Joseph

PY - 2024/11/1

Y1 - 2024/11/1

N2 - 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 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).

AB - 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 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).

U2 - 10.1007/s11139-024-00944-0

DO - 10.1007/s11139-024-00944-0

M3 - Article (Academic Journal)

SN - 1382-4090

VL - 65

JO - Ramanujan Journal

JF - Ramanujan Journal

IS - 3

M1 - 1421–1447

ER -

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

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