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On the integral dilatations of the “fractional part” function. (Sur les dilatations entières de la fonction “partie fractionnaire”.) (French. English summary) Zbl 1196.11117
Summary: With $$e_n(t) =\{t/n\}$$ and $$\mathcal{H}: = L^2(0,+\infty; t^{-2}\,dt)$$, we prove $\frac{6}{5} + O(n^{-1}) \leq n^2 \cdot \text{dist}_{\mathcal{H}}^2 \bigl (e_n, \text{Vect}(e_1, \dots, e_{n-1})\bigr) \leq \frac{3}{4} \log n + O(1).$
See the article of L. Báez-Duarte, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 14, No. 1, 5–11 (2003; Zbl 1097.11041).
##### MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses
##### Keywords:
Hilbert space; fractional part
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##### References:
 [1] L. Báez-Duarte, A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis , Rend. Mat. Acc. Lincei (9), 14 (2003), 5–11. · Zbl 1097.11041 [2] L. Báez-Duarte, M. Balazard, B. Landreau et E. Saias, Notes sur la fonction $$\zeta$$ de Riemann, 3 , Advances in Mathematics 149 (2000), 130–144. · Zbl 1008.11032 [3] L. Báez-Duarte, M. Balazard, B. Landreau et E. Saias, Étude de l’autocorrélation multiplicative de la fonction \og partie fractionnaire\fg , Ramanujan Journal 9 (2005), 215–240. · Zbl 1173.11343 [4] V. Vasyunin, On a biorthogonal system associated with the Riemann hypothesis , St-Petersburg Math. J. 7 (1996), 405–419.
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