Möbius inversion

In my last post we saw that \mu \ast \mathbf{1} = \varepsilon, that is, the Möbius function \mu is the inverse of \mathbf{1} with respect to Dirichlet convolution. This directly leads to an interesting principle called Möbius inversion.

Möbius inversion. Suppose g(n) is defined for n \geq 1 as the sum of another function f over all the divisors of n:

\displaystyle g(n) = \sum_{d \mid n} f(d).

Then we can “invert” this relationship to express f as a sum over g:

\displaystyle f(n) = \sum_{d \mid n} \mu(d) g(n/d).

The above is the “traditional” way of stating the principle, but I like to write it in this equivalent, more symmetric way:

\displaystyle f(n) = \sum_{ab = n} \mu(a) g(b)

where, as usual, the sum is over all factorizations of n into a product ab. This is the same as the previous equation, because ab = n if and only if a \mid n, in which case b = n/a. But now it becomes more clear that we are dealing with a Dirichlet convolution. Can you see how to prove this?

Proof. If we think in terms of Dirichlet convolution, the proof is almost trivial. Note first that saying g(n) = \sum_{d \mid n} f(n) is the same as saying g = \mathbf{1} \ast f (recall that taking the Dirichlet convolution with \mathbf{1} is the same as summing over all divisors). We can then take the Dirichlet convolution of both sides with \mu:

\displaystyle \begin{array}{rcl} \mu \ast g &=& \mu \ast (\mathbf{1} \ast f) \\ &=& (\mu \ast \mathbf{1}) \ast f \\ &=& \varepsilon \ast f \\ &=& f.\end{array}

In other words, Möbius inversion is really just saying that if f = \mathbf{1} \ast g, then \mu \ast f = g, because \mu is the inverse of \mathbf{1}.

Let’s try an example! Recall that \varphi denotes the Euler totient function, which counts how many positive integers less than or equal to n are relatively prime to n. For example, \varphi(5) = 4 since all the positive integers less than or equal to 5 (other than 5 itself) are relatively prime to it, and more generally \varphi(p) = p-1 for any prime p. As another example, \varphi(12) = 4 since the only positive integers between 1 and 11 which are relative prime to 12 are 1, 5, 7, and 11. In a previous post we considered a visual proof that

\displaystyle \sum_{d \mid n} \varphi(d) = n:

For example, as seen in the picture above, 12 = \varphi(1) + \varphi(2) + \varphi(3) + \varphi(4) + \varphi(6) + \varphi(12) = 1 + 1 + 2 + 2 + 2 + 4.

Now, this is in the perfect format to apply Möbius inversion: the sum over all divisors of the function g(n) = \varphi(n) is equal to the function f(n) = n. So we conclude that

\displaystyle \varphi(n) = \sum_{d \mid n} \mu(d) (n/d)

or, factoring out n,

\displaystyle \varphi(n) = n \sum_{d \mid n} \frac{\mu(d)}{d}

which is interesting since it expresses \varphi(n) as a fraction of n.

Let’s use this to compute \varphi(360). We have

\displaystyle \varphi(360) = \sum_{d \mid 360} \mu(d) (360/d).

The prime factorization of 360 is 2^3 \cdot 3^2 \cdot 5. We know that \mu(d) will be 0 for any divisor with any repeated factors, so those all cancel out; we only need to consider divisors which correspond to subsets of \{2,3,5\}. (n = 360 is the same example we used in our proof of the key property of the Möbius function.) So

\displaystyle \begin{array}{rcl} \varphi(360) &=& \mu(1) \cdot 360 \\ &+& \mu(2) \cdot 180 \\ &+& \mu(3) \cdot 120 \\ &+& \mu(5) \cdot 72 \\ &+& \mu(2\cdot 3) \cdot 60 \\ &+& \mu(2 \cdot 5) \cdot 36 \\ &+& \mu(3 \cdot 5) \cdot 24 \\ &+& \mu(2 \cdot 3 \cdot 5) \cdot 12 \end{array}

Remembering that \mu yields 1 for an even number of prime factors and -1 for an odd number, we can evaluate this as

\displaystyle \varphi(360) = 360 - 180 - 120 - 72 + 60 + 36 + 24 - 12 = 96.

(and we can use e.g. a computer to verify that this is correct). Note how this works: we start with 360 and then subtract out all the numbers divisible by 2 (all 180 of them), as well as those divisible by 3 (120) and 5 (72). But now we have subtracted some numbers multiple times, e.g. those which are divisible by both 2 and 3. So we add back in the numbers which are divisible by 2 \cdot 3 (60 of them), by 2 \cdot 5 (36), and by 3 \cdot 5 (24). Finally, we’ve now overcounted numbers which are divisible by all three, so we subtract off the 12 numbers which are divisible by 2 \cdot 3 \cdot 5. Egad! This seems really similar to PIE! Indeed, this is no accident: it turns out that the Principle of Inclusion-Exclusion is really just another kind of Möbius inversion. But that’s a subject for another post!

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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3 Responses to Möbius inversion

  1. sn0wleopard says:

    Ah, Möbius inversion is so much easier to understand via Dirichlet convolution!

    • Brent says:

      I know, right!? I now think it is pedagogically irresponsible to teach it any other way. But I had to work really hard and read a lot of different sources to put all of this together.

      • sn0wleopard says:

        Agreed! Thanks for putting all bits together. The whole series is great, please keep it going 🙂

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