A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis

Báez-Duarte, Luis

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 14 (2003), p. 5-11 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

According to the well-known Nyman-Beurling criterion the Riemann hypothesis is equivalent to the possibility of approximating the characteristic function of the interval $(0,1]$ in mean square norm by linear combinations of the dilations of the fractional parts $\{1/ax\}$ for real $a$ greater than $1$. It was conjectured and established here that the statement remains true if the dilations are restricted to those where the $a$’s are positive integers. A constructive sequence of such approximations is given.

Résumé

Il noto criterio di Nyman-Beurling per la validità dell’ipotesi di Riemann è equivalente alla possibilità di approssimare la funzione caratteristica dell’intervallo $(0,1]$ in media quadratica per mezzo di combinazioni lineari delle dilatazioni delle parti frazionarie $\{1/ax\}$ con $a$ reale e maggiore di $1$. In questa Nota si stabilisce la validità della congettura che il criterio resta vero se le dilatazioni sono ristrette a quelle dove il parametro $a$ è un intero. Si dà inoltre una costruzione esplicita di tali approssimazioni.

Détail

Publié le : 2003-03-01

MR 2057270 | Zbl 1097.11041

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