Explicit L 2 bounds for the Riemann ζ function
Daniele Dona1; Harald A. Helfgott1; Sebastian Zuniga Alterman2
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.

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≤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≤σ≤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≤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≤σ≤1.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1194
Classification: 11M06
Keywords: Riemann zeta function, $L^2$ norm, mean square bounds, explicit bounds, mean value theorem.
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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     title = {Explicit $L^2$ bounds for the {Riemann} $\zeta $ function},
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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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