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Suppose we have a function $f:\mathbb{N}\mapsto \mathbb{R}$, and we let $\Xi_{S}=\sum_{n\in S}f(n)$. Suppose further that for some known function $\mu:\mathbb{N}\mapsto \mathbb{R}$, $f(kn)=\mu(k)f(n)$ for all $k,n\in \mathbb{N}$. Is knowing $\Xi_\mathbb{N}$ enough to determine $\Xi_{C_l(m)}$ for all $l,m \in \mathbb{N}$, where $C_l(m)=\{n \in \mathbb{N}:n\equiv m \mod l\}$?
Obviously, we can determine $\Xi_{C_l(0)}=\mu(l)\Xi_{\mathbb{N}}$, as well as $\Xi_{C_2(1)}=\Xi_{\mathbb{N}}(1-\mu(2))$, but can we determine $\Xi$ for all or indeed any other congruence classes?

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Unfortunately, in the case of $f(x) = \mu(x) = \frac{1}{x^2}$, the sum $$\sum_{n \equiv 1 \, \bmod 4} \frac{1}{n^2}$$ is not even known to be irrational. As far as I know, no $\sum_{n \equiv 1 \, \bmod N} \frac{1}{n^2}$ has been proven irrational for $N \ge 3$.

Certainly we can't determine these numbers from $\frac{\pi^2}{6}$ in any reasonable sense.

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