Geometric sequences and zero-free region of the zeta function

Jongho Yang

doi:10.1016/j.crma.2017.11.021

Number theory

Geometric sequences and zero-free region of the zeta function

Presented by: the Editorial Board

Jongho Yang ¹

¹ Department of Mathematics, Korea University, Seoul 02841, Republic of Korea

Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 133-137.

Abstract (Original language)
Résumé

Let $N$ be the linear space of functions $\sum_{k = 1}^{n} a_{k} ρ (θ_{k} / x)$ with a condition $\sum_{k = 1}^{n} a_{k} θ_{k} = 0$ for $0 < θ_{k} \leq 1$ . Here $ρ (x)$ denotes the fractional part of x. Beurling pointed out that the problem of how well a constant function can be approximated by functions in $N$ is closely related to the zero-free region of the Riemann zeta function. More precisely, Báez-Duarte gave a zero-free region related to a $L^{p}$ -norm estimation of a constant function by using the Dirichlet series for the zeta function. In this paper, we consider the $L^{\infty}$ -norm estimation of a constant function and give a wider zero-free region than that of the Báez-Duarte result.

Soit $N$ l'espace vectoriel de fonctions $\sum_{k = 1}^{n} a_{k} ρ (θ_{k} / x)$ satisfaisant la condition $\sum_{k = 1}^{n} a_{k} θ_{k} = 0$ pour $0 < θ_{k} \leq 1$ , où $ρ (x)$ désigne la partie fractionnaire de x. Beurling a indiqué que le problème d'approximation d'une fonction constante par fonctions dans $N$ est étroitement lié à la région sans zéro de la fonction zêta de Riemann. Plus précisement, Báez-Duarte a donné une région sans zéro liée à une estimation de la norme $L^{p}$ d'une fonction constante en utilisant les séries de Dirichlet pour la fonction zêta. Dans cet article, nous considerons une estimation de la norme $L^{\infty}$ d'une fonction constante et donnons une région sans zéro plus large que celle du résultat de Báez-Duarte.

Received: 2017-08-18
Accepted: 2017-11-20
Published online: 2018-01-05

DOI: 10.1016/j.crma.2017.11.021

Author's affiliations:

Jongho Yang ¹

¹ Department of Mathematics, Korea University, Seoul 02841, Republic of Korea

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     title = {Geometric sequences and zero-free region of the zeta function},
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     doi = {10.1016/j.crma.2017.11.021},
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Jongho Yang. Geometric sequences and zero-free region of the zeta function. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 133-137. doi : 10.1016/j.crma.2017.11.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.11.021/

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