# Mobius function

The **Möbius function** is a multiplicative number theoretic function defined as follows:
Hence, we see that , and .

## Contents

## Multiplicity of the Function

One of the most important results of the Möbius function is that it is multiplicative, or that .

*Proof.* Let denote the number of integers not greater than and divisible by . If and are coprime then denotes the number of integers not greater than and divisible by and . Via a combinatorial argument using the Principle of Inclusion-Exclusion (PIE), it can be seen that the number of integers less than or equal to and not divisible by any one of a coprime set of integers is

and by taking the coprime set of integers to be the different (and distinct) prime factors of we get

Now the proof of the multiplicity of comes naturally from its definition and what we previously established. We see that

completing the proof

## Sums of Divisors

While the multiplicity of the Möbius function is its most crucial component, there are other relationships that we can draw from as well. One of the most notable is the following:

*Proof.*If where then

To finish, we simply can use the fact that , completing the proof. By a similar argument, one can also prove that

where is the number of distinct prime factors of

## Implications to Other Functions

Let be a multiplicative function with respect to . If there is such an , then

is also multiplicative.

*Proof.* If, for some , that , and then and runs through all divisors of . This means we have

which completes the proof. On a different note, this also provides an alternative proof for the theorem we proved earlier. By letting

the result follows by taking and for

## Applications

One of the major uses of the Möbius function is in the Mobius inversion formula. It also has ties to the Riemann zeta function, mainly that for ,

This comes from a couple of theorems. Firstly, if then

Secondly, if is an arithmetic function such that and is multiplicative such that is convergent, then

Our theorem follows by using the multiplicity of lastly. Doing so gives

which is what we wanted. The reason why this is important is because we can do more things by using this as a baseline. For example, it follows trivially that for