Q.Find $\mu(12)$ and $\mu(35)$.

The Möbius Function: From Intuition to Definition

The Möbius function is a tool from number theory that answers a surprisingly simple question: given a positive integer, how many distinct prime factors does it have, and are any of those factors repeated? It assigns each integer a value of $-1$, $0$, or $1$ based on the answer.

Think of it as a "prime factor signature" that cares only about two things: whether the number is square-free (no prime squared divides it), and if so, whether the number of distinct primes is odd or even.

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The Intuition First

Take any positive integer $n$. Factor it completely into primes:

Now ask two yes/no questions:

1. Is any exponent $a_i \ge 2$? That is, does any prime appear more than once in the factorization? If yes, the number is not square-free.
2. If the number is square-free, is the count $k$ of distinct primes odd or even?

The Möbius function $\mu(n)$ encodes the answers:

• If any prime repeats ($a_i \ge 2$ for some $i$), then $\mu(n) = 0$.
• If all primes appear exactly once ($a_i = 1$ for all $i$), then $\mu(n) = (-1)^k$, where $k$ is the number of distinct primes.

So $\mu(n) = 1$ when $n$ is square-free with an even number of prime factors, and $\mu(n) = -1$ when $n$ is square-free with an odd number of prime factors.

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The Special Case: $n = 1$

The number $1$ has no prime factors at all. It is square-free by convention, and the number of distinct primes is $k = 0$. Since $(-1)^0 = 1$, we define:

This is not arbitrary — it makes the Möbius function work beautifully in sums and inversions, as you will see later.

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The Precise Definition

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Worked Examples