Polynomial approximations in a generalized Nyman–Beurling criterion

Alouges, François; Darses, Sébastien; Hillion, Erwan

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 767-785.

Le critère de Nyman–Beurling, équivalent à l’hypothèse de Riemann (HR), est un problème d’approximation dans l’espace des fonctions de carré intégrable sur $(0,\infty )$, par des dilatations de facteurs ${\theta }_k\in (0,1)$, $k\ge 1$, de la fonction partie fractionnaire. Prendre les ${\theta }_k$ alétoires génère de nouvelles structures et de nouveaux critères. L’un d’eux est une condition suffisante pour HR qui revient à

• (i) montrer que la fonction indicatrice peut être approximée par des convolutions de la partie fractionnaire, et
• (ii) avoir un contrôle des coefficients de l’approximation.

Ce papier généralise les conditions (i) et (ii) afin d’obtenir un critère impliquant $\zeta (\sigma +it)\ne 0$ dans une bande $1/2<\sigma \le {\sigma }_0<1$. On identifie ensuite des fonctions pour lesquelles (i) est vérifiée inconditionnellement, grâce à des approximations polynomiales. Cela fournit, au passage, une courte preuve probabiliste d’une conséquence connue d’un théorème Taubérien. Dans ce contexte, la difficulté à prouver HR se reporte sur (ii), qui pourrait nécessiter une étude fine des matrices de Gram correspondantes. Nous obtenons deux structures remarquables de ces matrices. Nous montrons qu’un choix particulier des suites approximantes fournit une simplification remarquable de la matrice de Gram qui s’écrit alors sous la forme de matrices de Hankel par blocs.

The Nyman–Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on $(0,\infty )$, involving dilations of the fractional part function by factors ${\theta }_k\in (0,1)$, $k\ge 1$. Randomizing the ${\theta }_k$ generates new structures and criteria. One of them is a sufficient condition for RH that splits into

• (i) showing that the indicator function can be approximated by convolution with the fractional part,
• (ii) a control on the coefficients of the approximation.

This self-contained paper generalizes conditions (i) and (ii) that involve a ${\sigma }_0\in (1/2,1)$, and imply $\zeta (\sigma +it)\ne 0$ in the strip $1/2<\sigma \le {\sigma }_0<1$. We then identify functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener’s Tauberian theorem. In this context, the difficulty for proving RH is then reallocated in (ii), which heavily relies on the corresponding Gram matrices, for which two remarkable structures are obtained. We show that a particular tuning of the approximating sequence leads to a striking simplification of the second Gram matrix, then reading as a block Hankel form.

Classification : 41A30, 46E20, 60E05, 11M26
Mots clés : Nyman–Beurling criterion, Riemann Zeta function, Polynomial approximation, Probability distribution

[1] Báez-Duarte, Luis A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 14 (2003) no. 1, pp. 5-11 | Zbl

[2] Báez-Duarte, Luis; Balazard, Michel; Landreau, Bernard; Saias, Eric Étude de l’autocorrélation multiplicative de la fonction “partie fractionnaire”, Ramanujan J., Volume 9 (2005) no. 1, pp. 5-11 | Zbl

[3] Balazard, Michel Completeness problems and the Riemann hypothesis: an annotated bibliography, Number theory for the millennium I (Urbana, IL, 2000), A K Peters, 2000, pp. 21-48 | Zbl

[4] Bettin, Sandro; Conrey, Brian Period functions and cotangent sums, Algebra Number Theory, Volume 7 (2013) no. 1, pp. 215-242 | DOI | Zbl

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

[6] Borichev, Alexander On the closure of polynomials in weighted spaces of functions on the real line, Indiana Univ. Math. J., Volume 50 (2001) no. 2, pp. 829-846 | Zbl

[7] Borichev, Alexander Personal communication to Sébastien Darses, 2020

[8] Charlier, Christophe Asymptotics of Hankel determinants with a one-cut regular potential and Fisher-Hartwig singularities, Int. Math. Res. Not., Volume 2019 (2019) no. 24, pp. 7515-7576 | DOI | Zbl

[9] Darses, Sébastien; Hillion, Erwan An exponentially-averaged Vasyunin formula, Proc. Am. Math. Soc., Volume 149 (2021) no. 7, pp. 2969-2982 | DOI | Zbl

[10] Darses, Sébastien; Hillion, Erwan On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function, Confluentes Math., Volume 13 (2021) no. 1, pp. 43-59 | DOI | Zbl

[11] Ingham, Albert E. A note on Fourier transforms, J. Lond. Math. Soc., Volume 9 (1934), pp. 29-32 | DOI | Zbl

[12] Krasovsky, Igor V. Correlations of the characteristic polynomials in the Gaussian unitary ensemble or a singular Hankel determinant, Duke Math. J., Volume 139 (2007) no. 3, pp. 581-619 | Zbl

[13] Maier, Helmut; Rassias, Michael Th. The order of magnitude for moments for certain cotangent sums, J. Math. Anal. Appl., Volume 429 (2015) no. 1, pp. 576-590 | DOI | Zbl

[14] Maier, Helmut; Rassias, Michael Th. Generalizations of a cotangent sum associated to the Estermann zeta function, Commun. Contemp. Math., Volume 18 (2016) no. 1, 1550078, 89 pages | Zbl

[15] Mergelyan, Sergeĭ N. Weighted approximation by polynomials, Usp. Mat. Nauk, Volume 11 (1956), pp. 107-152

[16] Nikolski, Nikolaĭ K. Operators, functions, and systems: an easy reading. Volume 2: Model operators and systems, Mathematical Surveys and Monographs, 93, American Mathematical Society, 2002

[17] Nyman, Bertil On the one-dimensional translation group and semi-group in certain function spaces, Ph. D. Thesis, University of Uppsala (Sweden) (1950)

[18] Swaenepoel, Cathy Digits of prime numbers and other remarkable sequences, Ph. D. Thesis, Université d’Aix-Marseille (France) (2019)

[19] Tenenbaum, Gérald Introduction à la théorie analytique et probabiliste des nombres, Contributions in Mathematical and Computational Sciences, 1, Société Mathématique de France, 1995 | Zbl

[20] Titchmarsh, Edward C. The theory of the Riemann zeta-function, Oxford Science Publications, Clarendon Press, 1986

[21] Vasyunin, Vasiliĭ I. On a biorthogonal system related with the Riemann hypothesis, Algebra Anal., Volume 7 (1995) no. 3, pp. 118-135 | Zbl