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 , par des dilatations de facteurs , , de la fonction partie fractionnaire. Prendre les 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 dans une bande . 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 , involving dilations of the fractional part function by factors , . Randomizing the 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 , and imply in the strip . 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.
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Mots clés : Nyman–Beurling criterion, Riemann Zeta function, Polynomial approximation, Probability distribution
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@article{JTNB_2022__34_3_767_0,
author = {Fran\c{c}ois Alouges and S\'ebastien Darses and Erwan Hillion},
title = {Polynomial approximations in a generalized {Nyman{\textendash}Beurling} criterion},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {767--785},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {34},
number = {3},
year = {2022},
doi = {10.5802/jtnb.1227},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1227/}
}
TY - JOUR AU - François Alouges AU - Sébastien Darses AU - Erwan Hillion TI - Polynomial approximations in a generalized Nyman–Beurling criterion JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 767 EP - 785 VL - 34 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1227/ DO - 10.5802/jtnb.1227 LA - en ID - JTNB_2022__34_3_767_0 ER -
%0 Journal Article %A François Alouges %A Sébastien Darses %A Erwan Hillion %T Polynomial approximations in a generalized Nyman–Beurling criterion %J Journal de théorie des nombres de Bordeaux %D 2022 %P 767-785 %V 34 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1227/ %R 10.5802/jtnb.1227 %G en %F JTNB_2022__34_3_767_0
François Alouges; Sébastien Darses; Erwan Hillion. Polynomial approximations in a generalized Nyman–Beurling criterion. Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 767-785. doi : 10.5802/jtnb.1227. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1227/
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