On a question of Balazard and Saias related to the Riemann hypothesis (Q861061)

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.