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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
##### Keywords:
Riemann hypothesis; orthogonal projection
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##### References:
  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  Balazard, M.; Saias, E., Notes sur la fonction ζ de Riemann, 1, Adv. math., 139, 310-321, (1998) · Zbl 0920.11062  Balazard, M.; Saias, E., Notes sur la fonction ζ de Riemann, 4, Adv. math., 188, 69-86, (2004) · Zbl 1096.11032  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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