On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function

Sébastien Darses; Erwan Hillion

doi:10.5802/cml.71

Sébastien Darses ¹; Erwan Hillion ¹

¹ Aix-Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France

Confluentes Mathematici, Volume 13 (2021) no. 1, pp. 43-59.

Abstract

The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on $(0, \infty)$ , which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors $θ_{k} \in (0, 1)$ , $k \geq 1$ . We develop probabilistic extensions of the Nyman-Beurling criterion by considering these $θ_{k}$ as random: this yields new structures and criteria, one of them having a significant overlap with the general strong Báez-Duarte criterion.

The main goal of the present paper is the study of the interplay between these probabilistic Nyman-Beurling criteria and the Riemann hypothesis. We are able to obtain equivalences in two main classes of examples: dilated structures as exponential $ℰ (𝓀)$ distributions, and random variables $Z_{k, n}$ , $1 \leq k \leq n$ , concentrated around $1 / k$ as $n$ is growing. By means of our probabilistic point of view, we bring an answer to a question raised by Báez-Duarte in 2005: the price to pay to consider non compactly supported kernels is a controlled condition on the coefficients of the involved approximations.

Received: 2019-12-18
Revised: 2021-05-14
Accepted: 2021-05-25
Published online: 2021-10-20

DOI: 10.5802/cml.71
Classification: 41A30, 46E20, 60E05, 11M26
Keywords: Number theory; Probability; Zeta function; Nyman-Beurling criterion; Báez-Duarte criterion
Author's affiliations:

Sébastien Darses ¹; Erwan Hillion ¹

¹ Aix-Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France

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Sébastien Darses; Erwan Hillion. On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function. Confluentes Mathematici, Volume 13 (2021) no. 1, pp. 43-59. doi : 10.5802/cml.71. https://cml.centre-mersenne.org/articles/10.5802/cml.71/

References
Cited by

[1] N. Alon, J.H. Spencer. The probabilistic method. Third edition. With an appendix on the life and work of Paul Erdös. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., Hoboken, NJ, 2008. | Zbl: 1148.05001

[2] L. Báez-Duarte. A class of invariant unitary operators. Adv. Math., 144 (1999), no. 1, 1–12. | Article | MR: 1692568 | Zbl: 0978.47025

[3] L. Báez-Duarte. A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Rend. Mat. Ac. Lincei, S. 9, 14 (2003) 1, 5-11. | Zbl: 1097.11041

[4] L. Báez-Duarte. A general strong Nyman-Beurling criterion for the Riemann hypothesis. Publications de l’Institut Mathématique, Nouvelle Série, 78 (2005), pp. 117–125. | Article | MR: 2218310 | Zbl: 1119.11048

[5] L. Báez-Duarte, M. Balazard, B. Landreau and É. Saias. Notes sur la fonction $ζ$ de Riemann, 3. (French) [Notes on the Riemann $ζ$ -function, 3] Adv. Math., 149 (2000), no. 1, 130–144. | Article | MR: 1742356 | Zbl: 1008.11032

[6] L. Báez-Duarte, M. Balazard, B. Landreau and É. Saias. Étude de l’autocorrélation multiplicative de la fonction “partie fractionnaire”. (French) The Ramanujan Journal, 9(1) (2005), pp. 215–240. | Article | Zbl: 1173.11343

[7] L. Báez-Duarte, M. Balazard, B. Landreau and É. Saias. Document de travail – Étude de l’autocorrélation multiplicative de la fonction “partie fractionnaire”. (French)

[8] M. Balazard. Un siècle et demi de recherches sur l’hypothèse de Riemann. La Gazette des mathématiques, 126 (2010), pp.7–24. | Zbl: 1298.11087

[9] M. Balazard and A. de Roton. Sur un critère de Báez-Duarte pour l’hypothèse de Riemann. International Journal of Number Theory, 6(04) (2010), pp. 883–903. | Article | Zbl: 1201.11088

[10] M. Balazard, and É. Saias. Notes sur la fonction $ζ$ de Riemann, 4. Advances in Mathematics, 188(1) (2004), pp. 69–86. | Article | MR: 2083093 | Zbl: 1096.11032

[11] A. Beurling. A closure problem related to the Riemann Zeta-function. Proceedings of the National Academy of Sciences, 41(5) (1955), pp. 312–314. | Article | MR: 70655 | Zbl: 0065.30303

[12] J.F. Burnol. A lower bound in an approximation problem involving the zeros of the Riemann zeta function. Advances in Mathematics, 170(1) (2002), pp.56–70. | Article | MR: 1929303 | Zbl: 1029.11045

[13] J.F. Burnol. Entrelacement de co-Poisson. (French) [Co-Poisson links] Ann. Inst. Fourier (Grenoble) 57 (2007), no. 2, 525–602. | Article | MR: 2310951 | Zbl: 1177.11074

[14] J.B. Conrey. The Riemann hypothesis. Notices Amer. Math. Soc., 50 (2003), no. 3, 341–353. | Zbl: 1160.11341

[15] S. Darses and E. Hillion. An exponentially-averaged Vasyunin formula. Proc. of the American Math. Soc. To appear https://doi.org/10.1090/proc/15422. | Article | MR: 4257808 | Zbl: 07352296

[16] C. Delaunay, E. Fricain, E. Mosaki, O. Robert. Zero-free regions for Dirichlet series. Trans. Amer. Math. Soc., 365 (2013), no. 6, 3227–3253. | Article | MR: 3034464 | Zbl: 1322.11091

[17] N.Nikolski. Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann $ζ$ -function. Annales de l’institut Fourier Vol. 45, No. 1 (1995), pp. 143-159. | Article | MR: 1324128 | Zbl: 0816.30026

[18] B. Nyman. On the one-dimensional translation group and semi-group in certain function spaces. Thesis, University of Uppsala, 1950. | Zbl: 0037.35401

[19] G. Tenenbaum. Introduction à la théorie analytique et probabiliste des nombres, Société mathématique de France, 1995. | Zbl: 0880.11001

[20] E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press Oxford University Press, New York, 1986. | Zbl: 0601.10026

[21] V.I. Vasyunin. On a biorthogonal system associated with the Riemann hypothesis. (Russian) Algebra i Analiz 7, no. 3 (1995): 118-35; translation in St. Petersburg Mathematical Journal 7, no. 3 (1996): 405-19.

[22] A. Weingartner. On a question of Balazard and Saias related to the Riemann hypothesis. Adv. Math. 208 (2007), no. 2, 905–908. | Article | MR: 2304340 | Zbl: 1121.11058

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