On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function
Confluentes Mathematici, Volume 13 (2021) no. 1, pp. 43-59.

The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on (0,), which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors θ k (0,1), k1. 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 (𝓀) distributions, and random variables Z k,n , 1kn, 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.

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

[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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