Explicit $L^2$ bounds for the Riemann $\zeta $ function

Daniele Dona; Harald A. Helfgott; Sebastian Zuniga Alterman

doi:10.5802/jtnb.1194

Explicit

L^{2}

bounds for the Riemann

ζ

function

Daniele Dona ¹; Harald A. Helfgott ¹; Sebastian Zuniga Alterman ²

¹ Mathematisches Institut Georg-August-Universität Göttingen Bunsenstraße 3-5 37073 Göttingen, Germany
² Institut de Mathématiques de Jussieu Université Paris Diderot P7, Bâtiment Sophie Germain 8 Place Aurélie Nemours 75013 Paris, France

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 91-133.

Abstract
Résumé

Explicit bounds on the tails of the zeta function $ζ$ are needed for applications, notably for integrals involving $ζ$ on vertical lines or other paths going to infinity. Here we bound weighted $L^{2}$ norms of tails of $ζ$ .

Two approaches are followed, each giving the better result on a different range. The first one is inspired by the proof of the standard mean value theorem for Dirichlet polynomials. The second approach, superior for large $T$ , is based on classical lines, starting with an approximation to $ζ$ via Euler–Maclaurin.

Both bounds give main terms of the correct order for $0 < σ \leq 1$ and are strong enough to be of practical use for the rigorous computation of improper integrals.

We also present bounds for the $L^{2}$ norm of $ζ$ in $[1, T]$ for $0 \leq σ \leq 1$ .

Des bornes explicites pour la fonction zêta $ζ$ loin de la droite réelle sont nécessaires pour des applications, notamment aux intégrales de $ζ$ sur des lignes verticales ou bien sur d’autres chemins. Ici, nous bornons des normes $L^{2}$ ponderées de la fonction zêta loin de la droite réelle.

Nous suivons deux approches, chacune donnant le meilleur ré-sultat dans un certain rang. La première est inspirée par le théo-rème de la valeur moyenne pour les polynômes de Dirichlet. La deuxième, supérieure pour T grand, est basée sur des résultats classiques, en commençant par une approximation de $ζ$ via la formule d’Euler–Maclaurin.

Ces bornes donnent toutes les deux des termes principaux d’or-dre correct pour $0 < σ \leq 1$ . Elles sont assez fortes pour être d’utilité pratique dans le calcul numérique rigoureux d’intégrales impropres.

Nous présentons aussi des bornes pour la norme $L^{2}$ de $ζ$ dans $[1, T]$ pour $0 \leq σ \leq 1$ .

Received: 2020-08-15
Revised: 2021-09-24
Accepted: 2021-10-29
Published online: 2022-07-07

DOI: 10.5802/jtnb.1194

Classification: 11M06
Keywords: Riemann zeta function, $L^2$ norm, mean square bounds, explicit bounds, mean value theorem.

Author's affiliations:

Daniele Dona ¹; Harald A. Helfgott ¹; Sebastian Zuniga Alterman ²

¹ Mathematisches Institut Georg-August-Universität Göttingen Bunsenstraße 3-5 37073 Göttingen, Germany
² Institut de Mathématiques de Jussieu Université Paris Diderot P7, Bâtiment Sophie Germain 8 Place Aurélie Nemours 75013 Paris, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Daniele Dona and Harald A. Helfgott and Sebastian Zuniga Alterman},
     title = {Explicit $L^2$ bounds for the {Riemann} $\zeta $ function},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {91--133},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Daniele Dona; Harald A. Helfgott; Sebastian Zuniga Alterman. Explicit $L^2$ bounds for the Riemann $\zeta $ function. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 91-133. doi : 10.5802/jtnb.1194. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1194/

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