Number theory
Geometric sequences and zero-free region of the zeta function
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 133-137.

Let N be the linear space of functions k=1nakρ(θk/x) with a condition k=1nakθk=0 for 0<θk1. 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 Lp-norm estimation of a constant function by using the Dirichlet series for the zeta function. In this paper, we consider the L-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 k=1nakρ(θk/x) satisfaisant la condition k=1nakθk=0 pour 0<θk1, 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 Lp 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 d'une fonction constante et donnons une région sans zéro plus large que celle du résultat de Báez-Duarte.

Published online:
DOI: 10.1016/j.crma.2017.11.021
Yang, Jongho 1

1 Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
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Yang, Jongho. 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.

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

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

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

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

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

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

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

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

[10] Pintz, J. Oscillatory properties of M(x)=nxμ(n). III, Acta Arith., Volume 43 (1984), pp. 105-113

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

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

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