# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.