Applications of the mobius inversion in algebraic or arithmetic geometry

Question

I was wondering if there were any generalisations or applications of the ideas of the Möbius inversion formula to more modern areas of mathematics such as algebraic or arithmetic geometry.

I know they are useful when calculating the minimal polynomials of roots of unity with degree exactly $n$, however couldn't find anything deeper in that vein.

Answer 1

The Mobius inversion formula that you ask about says for real or complex-valued functions $f$ and $g$ on the positive integers that $$ f(n) = \sum_{d \mid n} g(d) \text{ for all } n \geq 1 \Longleftrightarrow g(n) = \sum_{d \mid n} \mu\left(\frac{n}{d}\right)f(d) \text{ for all } n \geq 1. $$ The proof of that equivalence depends on the formula $$ \sum_{d \mid n} \mu(d) = \begin{cases} 1, & {\rm if } \ n = 1, \\ 0, & {\rm if } \ n > 1. \end{cases} $$ which I'll call the fundamental identity for the Mobius function. This identity could be considered a recursive (and very unmotivated) definition of the Mobius function rather than a property of it. In any case, a result based on the fundamental identity could be regarded as a consequence of Mobius inversion.

Here are some ways Mobius inversion or the fundamental identity are used.

1. See p. 60 of Dwork's paper on the zeta-function of a hypersurface here.

2. The fundamental identity is equivalent to being able to write the exponential function as the following formal power series $$ e^x = \prod_{n \geq 1} (1 - x^n)^{-\mu(n)/n}. $$ Removing the terms at $n$ divisible by $p$ leaves us with the Artin-Hasse exponential, which is defined as $$ {\rm AH}_p(x) = \prod_{\stackrel{n \geq 1}{ p \nmid n}} (1 - x^n)^{-\mu(n)/n}, $$ which shows ${\rm AH}_p(x)$ has coefficients in $\mathbf Z_p$ since $\mu(n)/n \in \mathbf Z_p$ when $p \nmid n$.

3. The fundamental identity is equivalent to the Dirichlet series representation (for ${\rm Re}(s) > 1$) $$ \frac{1}{\zeta(s)} = \sum_{n \geq 1} \frac{\mu(n)}{n^s} $$ and also $$ \frac{1}{L(s,\chi)} = \sum_{n \geq 1} \frac{\mu(n)\chi(n)}{n^s} $$ for a Dirichlet character $\chi$. These are one of the reasons for the importance of the Mobius function in analytic number theory. And analytic number theory makes use of a second Mobius inversion formula: for two (real or complex-valued) functions $f$ and $g$ on $[1,\infty)$, $$ f(x) = \sum_{n \leq x} g(x/n) \text{ for all } x \geq 1 \Longleftrightarrow g(x) = \sum_{n \leq x} \mu(n)f(x/n) \text{ for all } x \geq 1. $$

4. Mobius inversion holds not just for functions from the positive integers to $\mathbf R$ or $\mathbf C$, but functions from the positive integers to any abelian group $G$. Writing the operation in $G$ multiplicatively instead of additively, for two functions $f, g : \mathbf Z^+ \to G$ $$ f(n) = \prod_{d \mid n} g(d) \text{ for all } n \geq 1 \Longleftrightarrow g(n) = \prod_{d \mid n} f(d)^{\mu(n/d)} \text{ for all } n \geq 1. $$ As an example, take $f(n) = x^n - 1$ and $g(n) = \Phi_n(x)$ (the $n$th cyclotomic polynomial) with $G = \mathbf C(x)^\times$. Then the formula $x^n - 1 = \prod_{d \mid n} \Phi_d(x)$ for all $n \geq 1$ implies by Mobius inversion for $\mathbf C(x)^\times$-valued functions that $$ \Phi_n(x) = \prod_{d \mid n} (x^d-1)^{\mu(n/d)}. $$ For $n > 1$, since $\sum_{d \mid n} \mu(n/d) = \sum_{d \mid n} \mu(d) = 0$ we can change the order of subtraction in all the factors on the right side: $$ \Phi_n(x) = \prod_{d \mid n} (1-x^d)^{\mu(n/d)} $$ for $n \geq 2$. The left side is in $\mathbf C[x]$ and the right side is in $\mathbf Z[[x]]$; since $\mathbf C[x] \cap \mathbf Z[[x]] = \mathbf Z[x]$, we obtain a curious proof by Mobius inversion that $\Phi_n(x) \in \mathbf Z[x]$ when $n \geq 2$ (it's obvious when $n = 1$).

5. The definition of the Mobius function in terms of values on squarefree and nonsquarefree positive integers extends naturally to a function on nonzero ideals in a Dedekind domain and this leads to a generalization of the zeta function identity in item 3 above to the zeta function of a number field: $$ \frac{1}{\zeta_K(s)} = \sum_{\mathfrak a} \frac{\mu_K(\mathfrak a)}{{\rm N}(\mathfrak a)^s} $$ for ${\rm Re}(s) > 1$. As in the case of positive integers, $\mu_K(\mathfrak a)$ is $0$ or $\pm 1$ depending on how $\mathfrak a$ factors into (nonzero) prime ideals.

6. The Mobius function generalizes from positive integers to other posets: look up the Mobius function of a poset, where it no longer has to have only values $\{0, 1, -1\}$.

7. Here is surprising application of Mobius inversion: it implies all perfect numbers are even. Indeed, if $n$ is perfect then $\sum_{d\mid n} d = 2n$, so by Mobius inversion $$ n = \sum_{d\mid n} \mu(n/d)2d = 2\sum_{d\mid n} \mu(n/d)d, $$ which is even. Of course this “proof” has a mistake. If you did not catch it on your first reading, then reread it until you find the error.