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
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/

References
Cited by

[1] L. Báez-Duarte On Beurling's real variable reformulation of the Riemann hypothesis, Adv. Math., Volume 101 (1993), pp. 10-30

[2] M. Balazard Completeness problem and the Riemann hypothesis: an annotated bibliography, Surveys in Number Theory: Papers from the Millennial Conference on Number Theory, CRC Press, Boca Raton, FL, USA, 2002, pp. 1-28

[3] M. Balazard; E. Saias The Nyman–Beurling equivalent form for the Riemann hypothesis, Expo. Math., Volume 18 (2000), pp. 131-138

[4] H. Bercovici; C. Foias A real variable restatement of Riemann's hypothesis, Isr. J. Math., Volume 48 (1984), pp. 56-68

[5] A. Beurling A closure problem related to the Riemann zeta-function, Proc. Natl. Acad. Sci. USA, Volume 41 (1955), pp. 312-314

[6] B. Green; T. Tao The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2), Volume 175 (2012), pp. 541-566

[7] N. Ng The distribution of the summatory function of the Möbius function, Proc. Lond. Math. Soc., Volume 89 (2004), pp. 361-389

[8] B. Nyman On the One-Dimensional Translation Group and Semi-group in Certain Function Spaces, University of Uppsala, Sweden, 1950 (PhD thesis)

[9] A.M. Odlyzko; H.J.J. te Riele Disproof of the Mertens conjecture, J. Reine Angew. Math., Volume 357 (1985), pp. 138-160

[10] J. Pintz Oscillatory properties of $M (x) = \sum_{n \leq x} μ (n)$ . III, Acta Arith., Volume 43 (1984), pp. 105-113

[11] J. Yang A note on Nyman–Beurling's approach to the Riemann hypothesis, Integral Equ. Oper. Theory, Volume 83 (2015), pp. 447-449

[12] J. Yang A generalization of Beurling's criterion for the Riemann hypothesis, J. Number Theory, Volume 164 (2016), pp. 299-302

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