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

Research output: Contribution to journal › Article (Academic Journal) › peer-review

5 Downloads (Pure)

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

Access to Document

10.1007/s11139-024-00944-0

Full-text PDF (accepted manuscript)
This is the accepted author manuscript (AAM) of the article which has been made Open Access under the University of Bristol's Scholarly Works Policy. The final published version (Version of Record) can be found on the publisher's website. The copyright of any third-party content, such as images, remains with the copyright holder.
Accepted author manuscript, 502 KBLicence: CC BY

https://link.springer.com/article/10.1007/s11139-024-00944-0

Handle.net

Persistent link

Cite this

@article{9b4d8a6c12954ae6a8d1a0aa53def4d7,

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).",

author = "S{\'e}bastien Darses and Joseph Najnudel",

year = "2024",

month = nov,

day = "1",

doi = "10.1007/s11139-024-00944-0",

language = "English",

volume = "65",

journal = "Ramanujan Journal",

issn = "1382-4090",

publisher = "Springer Verlag",

number = "3",

}

TY - JOUR

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

Abstract

Access to Document

Handle.net

Fingerprint

Cite this