Geodesic
A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 1, pp. 5-11.

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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.
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. 
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Báez-Duarte, Luis. A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 1, pp. 5-11. http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_1_a0/

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