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Questions tagged [mobius-inversion]

For questions related to Möbius inversion and its applications.

In number theory, the Möbius inversion formula states that if $f$ and $g$ are arithmetic functions for which

$$g(n) = \sum_{d | n} f(d)$$

for every integer $n \ge 1$, then we can recover $f$ by

$$f(n) = \sum_{d | n} \mu(d) g\left(\frac{n}{d}\right)$$

where $\mu$ is the Möbius function. Written in terms of Dirichlet convolutions,

$$g = f \ast 1 \implies f = \mu \ast g$$

Reference: Möbius inversion formula.

203 questions
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Combinatorial Interpretation of a Certain Product of Factorials

Let $\mu$ denote the Moebius function. What is a combinatorial interpretation of the following integer, \begin{align} \prod_{d \mid n} d!^{\,\mu(n/d)}, \end{align} where the product is taken over divisors of $n$? Does it have a simpler…
user02138
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How many ways to arrange $n$ points in $(\Bbb F_q)^2$ with no three collinear?

How many ways are there to arrange $n$ points in the finite field plane $(\Bbb F_q)^2$ with no three of the points collinear? An easy upper bound is $(q^2)^n=q^{2n}$, but of course it's less than that. (Of course, if I asked the same question over…
Akiva Weinberger
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Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Möbius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised Möbius inversion formula states that the above is…
Meow
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Möbius Inversion and Fourier transform.

Is there any relation between the Möbius Inversion formula for posets and the inversion theorem for the Fourier transform? The two formulas have a conceptual similarity in that each says that a convolution operator is inverse to another, which…
Arka
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Looking for help understanding the Möbius Inversion Formula

I was recently told that the Möbius Inversion Formula can be applied to the Chebyshev Function. Let $\vartheta(x)$,$\psi(x)$ be the first and second Chebyshev functions so that: $$\vartheta(x) = \sum_{p\le{x}}\log p$$ $$\psi(x) =…
Larry Freeman
  • 10,189
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2 answers

Understanding the proof of Möbius inversion formula

I am trying to understand one step in the proof of the Möbius inversion formula. The theorem is Let $f(n)$ and $g(n)$ be functions defined for every positive integer $n$ satisfying $$f(n) = \sum_{d|n}g(d)$$ Then, g satisfies…
urpi
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Seeking for a proof on the relation $ \varphi(n)={\displaystyle \sum_{d|n}}d\mu\left(\frac{n}{d}\right), $ between Euler totient and Möbius function

Can someone help me prove the relation $$ \varphi\left(n\right)=\sum_{d|n}d\mu\left(\frac{n}{d}\right), $$ where $\mu$ is the Möbius function defined by $$ \mu\left(n\right)=\begin{cases} 1 & \mbox{if }n=1\\ \left(-1\right)^{t} & \mbox{if }n\mbox{…
vantonio1992
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Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results about primes? $\text{Li}_s(x)$ is the polylogarithm,…
Ethan Splaver
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Möbius inversion on the partition lattice

For some $n \in \mathbb N$, let $(\Pi_n, \le)$ be the poset of partitions of the set $\{1, 2, \dots, n\}$, where two partitions $\pi, \rho \in \Pi_n$ have the relation $\pi \le \rho$ if $\pi$ is a refinement of $\rho$, i.e., if every block in $\pi$…
KDS
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Monic Irreducible Polynomials over Finite Field

Let $F=\mathbb{F}_{q}$ be a finite field (so $q=p^k$ for some prime $p$ and positive integer $k$), and let $\varphi(d)$ denote the number of monic irreducible polynomials of degree $d$ in $F[X]$. I'm supposed to show that $\displaystyle{\sum_{d \mid…
Joe Wells
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Intuition for Möbius function on a poset

I am attempting to learn about Möbius inversion in the context of partial order theory. However, I'm hitting a bit of a mental block when it comes to understanding the Möbius function, and I'm looking for a clearer understanding of the motivation…
N. Virgo
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Proving $P(n) =n^{\phi(n)} \prod\limits_{d \mid n} \left(\frac{d!}{d^d} \right)^{\mu(n/d)}$

Actually, I posted this long ago in MO but did not get a reply as it was unfit. Now this is an exercise in some textbook (I think Apostol), and I would be happy to receive some answers. Let $P(n)$ be the product of positive integers which are $\leq…
anonymous
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Continuous version of the Möbius inversion theorem

Is there a continuous version of Möbius Inversion. Essentially, using integrals instead of sums.
Mayank Pandey
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Count pairwise coprime triples such that the maximum number of the triple is not greater than N

Problem Statement: Given N you are to count the number of pairwise coprime triples which satisfy $1≤a,b,c≤N$. Example: For example N=3, valid triples are…
user1068604
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2 answers

Sum of primitive roots is congruent to $\mu(p-1)$ using Moebius inversion?

Wikipedia has the result that Gauss proved that for a prime number $p$ the sum of its primitive roots is congruent to $\mu(p − 1) \pmod{p}$ in Article 81. I read it, but is there a faster proof using Moebius inversion instead of case by case…
Dedede
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