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On a question of Balazard and Saias related to the Riemann hypothesis. (English) Zbl 1121.11058
Let \(H\) be the Hilbert space of sequences of complex numbers \(x=(x(1),x(2),x(3),\ldots)\) that satisfy \(\langle x,x\rangle<\infty\), where the inner product \(\langle x,y\rangle\) is given by \[ \langle x,y\rangle=\sum_{j=1}^\infty \frac{x(j)\overline{y(j)}}{j(j+1)}. \] For an integer \(k\geq 2\) define \(r_k\in H\) to be the sequence whose \(j\)th term \(r_k(j)\) is the remainder when \(j\) is divided by \(k\). Let \[ g_n:=\sum_{k=2}^nc_{n,k}r_k \] denote the orthogonal projection of \({\mathbf 1}:=(1,1,1,\ldots)\) onto the subspace of \(H\) generated by the finite set \(\{r_k:2\leq k\leq n\}.\) The author proves that the Riemann hypothesis is equivalent to \[ \lim_{n\rightarrow \infty}c_{n,k}=-\frac{\mu(k)}{k}\,\text{for all}\,k\geq 2, \] where \(\mu\) is the Möbius function.
MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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[1] Báez-Duarte, L., A strengthening of the nyman-Beurling criterion for the Riemann hypothesis, Atti accad. naz. lincei cl. sci. fis. mat. natur. rend. lincei (9) mat. appl., 14, 1, 5-11, (2003) · Zbl 1097.11041
[2] Balazard, M.; Saias, E., Notes sur la fonction ζ de Riemann, 1, Adv. math., 139, 310-321, (1998) · Zbl 0920.11062
[3] Balazard, M.; Saias, E., Notes sur la fonction ζ de Riemann, 4, Adv. math., 188, 69-86, (2004) · Zbl 1096.11032
[4] Vasyunin, V.I., On a biorthogonal system related with the Riemann hypothesis, Saint |St. Petersburg math. J., 7, 405-419, (1996)
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