Let $G_n$ be the finite group of square-free integers that are divisors of $p_n\#$ under the law $x \star y := \dfrac{ x y}{\gcd(x,y)^2}$. It obviously forms a boolean group. Where $p_n\#$ is $p_2 p_3 \cdots p_n$.
Now take the map:
$$ \chi(s):= \sum_{g \in G_n} \mu(d)(g \mid s) $$
where $(g \mid s) := 1$ if $g$ divides $s$ in the ring $\Bbb{Z}$; or otherwise $0$. So you could write this as a sum over $g \mid s$ (typical number theory style) but we're not going to here.
Then $\chi(s)$ indicates on the interval $[p_n + 1, p_{n+1}^2 - 1]$ whether $s$ is a prime number.
I'm wondering if $\chi(s)$ has any relation to the group $G_n$? Obviously it's a summation over $G_n$. Equivalently it indicates on the interval $[1, p_{n+1}^2 - 1]$ whether $s$ is a prime distinct from any $p_1, p_2, \dots, p_n$.
Thanks!
Not only does $\chi_n$ we'll call it indicate primality on $\pm [p_n + 1, \dots, p_{n+1}^2 - 1] := I_n$ but it also indicates the same on $\langle I_n\rangle^{\cdot}$ ie. the multiplicative monoid generated by numbers in that interval. $1 \in M_n := \langle I_n\rangle^{\cdot}$ is included because it simply always evaluates to $1 =: p_0$ one of our "primes". Just a note.
Next question is take $B = \{ \bigcap_{i\in I} M_{i} : I \subset \Bbb{N} \text{ is a finite set}\}$ i.e. the set of all finite intersections of these monoids as a basis for open sets. Then clearly these sets at least generate a topology on $\Bbb{Z}\setminus 0$.
So can we form a sheaf in which the $\chi_n$ is a restriction of $\chi_m$ for all $m \geq n$ for example on the interval $[p_m+1, p_{n+1}^2 -1]$. And the same thing happens on the monoid side. So I'm wondering if this topology where open sets are arbitrary unions of certain monoids is a good basis for some sort of sheaf!
Attempt. Let $F:O(X)^{\text{op}} \to \textbf{CRing}$ be the commutative ring-valued sheaf on the open sets described in the question post. So each open set $U \subset X$ is an arbitrary union of monoids generated by certain intervals. The union of all these monoids clearly equals the whole space $X = \Bbb{Z}\setminus 0$ because their generating sets the intervals do, by definition. Also the empty interval is in there by definition of finite union including $\varnothing$, etc. So it really is a topology. It's open sets are unions of the multiplicative monoids generated respectively by this list of $\pm$ intervals:
$\pm[2,3],\pm[3],\pm[3,7],\pm[4,7],\pm[4, 24], \dots$
The commutative ring:
$$ R = \{f:\Bbb{Z}\setminus 0 \to \Bbb{Z}\} $$
And let $F:U \mapsto \{ f\vert_U : U \to \Bbb{Z}, f\in R\}$, then each $F(U)$ is a commmutative ring as well.
Then the usual indicator function of primality on $\Bbb{Z}\setminus 0$ or $\chi_{\Bbb{P}} \in F(X)$ is the unique map in the definition of sheaf such that restricted to $M_i$ it induces $\chi_i$.