4.3: The Mobius Function and the Mobius Inversion Formula

We start by defining the Mobius function which investigates integers in terms of their prime decomposition. We then determine the Mobius inversion formula which determines the values of the a function at a given integer in terms of its summatory function.

\(\mu(n)=\left\{\right .\)

Note that if is divisible by a power of a prime higher than one then .

In connection with the above definition, we have the following

An integer is said to be square-free, if no square divides it, i.e. if there does not exist an integer such that .

It is immediate (prove as exercise) that the prime-number factorization of a square-free integer contains only distinct primes.

Notice that , , and .

We now prove that is a multiplicative function.

The Mobius function is multiplicative.

Let and be two relatively prime integers. We have to prove that If , then the equality holds. Also, without loss of generality, if , then the equality is also obvious. Now suppose that or is divisible by a power of prime higher than 1, then What remains to prove that if and are square-free integers say where are distinct primes and where . Since , then there are no common primes in the prime decomposition between and . Thus

In the following theorem, we prove that the summatory function of the Mobius function takes only the values or .

Let , then satisfies \[F(n)=\left\{\right .\]

For , we have . Let us now find for any integer . Notice that Thus by Theorem 36, for any integer we have,

We now define the Mobius inversion formula. The Mobius inversion formula expresses the values of in terms of its summatory function of .

Suppose that is an arithmetic function and suppose that is its summatory function, then for all positive integers we have

We have Notice that unless and thus . Consequently we get

A good example of a Mobius inversion formula would be the inversion of and . These two functions are the summatory functions of and respectively. Thus we get and

Exercises

1. Find , and .
2. Find the value of for each integer with .
3. Use the Mobius inversion formula and the identity to show that where is a prime and is a positive integer.

Contributors and Attributions

• Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.