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Let $0 \in \Bbb{N}$ and let $[\cdot ] = \lfloor \cdot \rfloor$.
When $c \mid d \mid y\#$, let $x_{c,d} = 1 \pmod c, = -1 \pmod{d/c}$ by CRT (it can be any solution for each $c \mid d$).
We know that:
$$
f: \Bbb{N} \to \Bbb{N} \\
f(z) = \sum_{d \mid \sqrt{z}\#}\mu(d) \sum_{2 \nmid c \mid d}\lfloo...
