Euler characteristic and the Mertens function?

Question

I am interested if there can be given any applications of this topology on prime powers and a sheaf on it, to number theoretic questions ( I am looking for known results in elementary number theory, which could be proven with this topology or sheaf on it. Thanks in advance for your help!):

Let $n \in \mathbb{N}_0$ be a natural number, which can be $=0$. We associate to this number the set:

\begin{equation}\label{eq1} S(n):=\{p^k : 1 \le k \le v_p(n), p\mid n , p \text{ is prime}\} \end{equation}

For instance we have $S(0)=\hat{\mathbb{P}}={}$the set of all prime powers ($1$ is not counted as a prime power). And $S(1) = \emptyset$.

The sets $S(n)$ with $n \in \mathbb{N}_0$ are the closed sets of a topology on the set of prime powers $\hat{\mathbb{P}}$, since we have:

\begin{equation}\label{eq2} S(m) \cap S(n)=S(\gcd(m,n)) \forall m,n \in \mathbb{N}_0, \end{equation}

\begin{equation}\label{eq3} S(m) \cup S(n)=S(\frac{mn}{\gcd(m,n)}) \forall m,n \in \mathbb{N}_0, \end{equation}

and by Bézout's theorem the $\gcd$ of an infinite number of natural numbers can be defined:

\begin{equation}\label{eq4} \bigcap_{i\in I} S(n_i) = S(\gcd(\{ n_i : i\in I\})). \end{equation}

We have

\begin{equation}\label{eq5} m = n \iff S(m) = S(n) \iff O(m) = O(n), \end{equation} where the open sets are defined as $O(n):=\hat{\mathbb{P}}-S(n)$.

So this allows us to think about natural numbers as the closed sets of a topological space on the prime powers.

We want to define a sheaf

\begin{equation}\label{eq6} F(O(n)):= n \cdot \mathbb{Z} \end{equation}

which to each open set $O(n)$ assigns the ideal, $n \cdot \mathbb{Z}$, which is an abelian group.

We have:

\begin{equation}\label{eq7} F(O(0))= F(\hat{\mathbb{P}}-S(0))=F(\emptyset) = 0 , F(O(1)) = F(\hat{\mathbb{P}}-S(1)) = F(\hat{\mathbb{P}}) = \mathbb{Z} \end{equation}

hence the global sections $F(\hat{\mathbb{P}})$ are the integers $\mathbb{Z}$ and the local sections $F(O(n))$, $n>0$ are the ideals viewed as abelian groups $n \mathbb{Z}$.

For $m\mid n$, $n>0$ the restriction morphisms $\rho(O(m),O(n))$ which will also be denoted as $\rho(m,n)$ are given by:

\begin{equation}\label{eq8} \rho(m,n): F(O(n)) \rightarrow F(O(m)), x \mapsto \frac{m}{n} \cdot x, \rho(m,n)(x) := \frac{m}{n} \cdot x. \end{equation}

For $m \in \mathbb{N}_0$, $n=0$ the restriction morphisms $\rho(m,0)=\rho(O(m),O(0))$ are given by:

\begin{equation}\label{eq9} \rho(m,n): F(O(n)) \rightarrow F(O(m)), x \mapsto 0, \rho(m,n)(x) := 0 \end{equation}

because $F(O(n)) = n \cdot \mathbb{Z} = 0$ and so each $x$ in $F(O(n))$ is equal to $0$ and can be mapped to $0$.

I have searched the literature but have not found anything related to this specific topology and sheaf on it. Using the subset topology, one can study these topologies from the point of view of finite topologies, which there is a lot of literature on it, but I have not yet delved deeper in this topic.

Disclaimer: This is a working draft, in case someone is interested in further properties of the topology, but I am more interested in applications of sheaf theory.

Second Edit: Sorry for the inconvenience. There was a bug in the construction of the simplicial complex, which was not intended to be so.

Here is a more concrete question: In A cell complex in number theory by Anders Björner, 2010 a number theoretic cell complex is described which has the property that the Euler characteristic is equal to the Mertens function:

$$M(n) = \sum_{k=1}^n \mu(k)$$

In the simplicial complex I am considering we might have:

$$|M(n)-1| =^? |\chi(n)|$$

where $\chi(n)$ denotes the Euler characteristic of the constructed simplicial complex.

I am using https://simplicial.readthedocs.io/en/latest/index.html for computing simplicial complexes:

from simplicial import SimplicialComplex
import itertools
def S(n):
    ll = []
    for p in prime_divisors(n):
        ll.extend([(p**k) for k in range(1,valuation(n,p)+1)])
    return ll
def one_simplex(cmplex, num):
    sn = S(num)
    # Hinzufügen von 0-Simplexen (Eckpunkten)
    bs = []
    for q in sn:
        if not cmplex.containsSimplex(q):
            s = cmplex.addSimplex(id=q)
        else:
            s = cmplex.closureOf(q)[0]
        bs.append(s)
    #print(bs)

    if len(bs)>1:
        cmplex.addSimplexWithBasis(bs = bs)

    return cmplex
Beispielverwendung der Funktion
for a in range(1,400):
    #c = SimplicialComplex()
    cmplex = SimplicialComplex()
    n = a
    rr = list(range(1,n+1))
    for d in rr:
        cmplex = one_simplex(cmplex,d)
    print("n = ",n)
    print("|Euler.characteristic(n)| = ", cmplex.eulerCharacteristic())
    print("|Mertens(n)-1| = ", abs(sum((moebius(x)) for x in rr)-1))
    print("|Mertens(n)-1| = |Euler.characteristic(n)| ?", abs(cmplex.eulerCharacteristic())== abs(sum((moebius(x)) for x in rr)-1))
    print("Betti-Numbers =", cmplex.bettiNumbers())

Here is the output data:

n =  1
|Euler.characteristic(n)| =  0
|Mertens(n)-1| =  0
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {}
n =  2
|Euler.characteristic(n)| =  1
|Mertens(n)-1| =  1
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 1}
n =  3
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 2}
n =  4
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 2, 1: 0}
n =  5
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 3, 1: 0}
n =  6
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 2, 1: 0}
n =  7
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 3, 1: 0}
n =  8
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 3, 1: 0, 2: 0}
n =  9
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 3, 1: 0, 2: 0}
n =  10
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 2, 1: 0, 2: 0}
n =  11
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 3, 1: 0, 2: 0}
n =  12
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 3, 1: 0, 2: 0}
n =  13
|Euler.characteristic(n)| =  4
|Mertens(n)-1| =  4
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 4, 1: 0, 2: 0}
n =  14
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 3, 1: 0, 2: 0}
n =  15
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 3, 1: 1, 2: 0}
n =  16
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 3, 1: 1, 2: 0, 3: 0}
n =  17
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 4, 1: 1, 2: 0, 3: 0}
n =  18
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 4, 1: 1, 2: 0, 3: 0}
n =  19
|Euler.characteristic(n)| =  4
|Mertens(n)-1| =  4
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 5, 1: 1, 2: 0, 3: 0}
n =  20
|Euler.characteristic(n)| =  4
|Mertens(n)-1| =  4
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 5, 1: 1, 2: 0, 3: 0}
n =  21
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 5, 1: 2, 2: 0, 3: 0}
n =  22
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 4, 1: 2, 2: 0, 3: 0}
n =  23
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 5, 1: 2, 2: 0, 3: 0}
n =  24
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 5, 1: 2, 2: 0, 3: 0}
n =  25
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 5, 1: 2, 2: 0, 3: 0}
n =  26
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 4, 1: 2, 2: 0, 3: 0}
n =  27
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 4, 1: 2, 2: 0, 3: 0}
n =  28
|Euler.characteristic(n)| =  2
|Mertens(n)-1| =  2
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 4, 1: 2, 2: 0, 3: 0}
n =  29
|Euler.characteristic(n)| =  3
|Mertens(n)-1| =  3
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 5, 1: 2, 2: 0, 3: 0}
n =  30
|Euler.characteristic(n)| =  4
|Mertens(n)-1| =  4
|Mertens(n)-1| = |Euler.characteristic(n)| ? True
Betti-Numbers = {0: 5, 1: 1, 2: 0, 3: 0}

Basically when we have any list of natural numbers $n_1,n_2,\cdots,n_r$ we can look at the closed sets: $S(n_1),S(n_2),\cdots,S(n_r)$

For each $S(n_i)$ I construct a simplex having as points the prime powers dividing $n_i$ from $S(n_i)$.

The simplicial complex of the code above is for the numbers $1,2,\cdots,n$.

The simplicial complex $\Delta_n$ is defined as:

$$\Delta_n:=\{T: T \subset S(k), 1 \le k \le n \}$$

The simplicial complex can be defined for each finite set of numbers $n_1,\cdots,n_r$:

$$\Delta(n_1,\cdots,n_r) := \{ T : T \subset S(n_k) , 1 \le k \le r \}$$