Arithmetic functions · Definition

Möbius function

elementary number theoryarithmetic functions

Before you start you need to know

The Möbius function is one of the most important arithmetic functions in number theory. It allows sums over divisors to be inverted through the inversion formula.

Definition

The Möbius function \(\mu(n)\) is an arithmetic function: it equals \(1\) if \(n=1\), \((-1)^k\) if \(n\) is a product of \(k\) distinct primes, and \(0\) if \(n\) has a repeated prime factor.

\[\sum_{d\mid n}\mu(d)=\begin{cases}1 & n=1\\ 0 & n>1\end{cases}\]

It is multiplicative: if \(\gcd(m,n)=1\), then \(\mu(mn)=\mu(m)\mu(n)\). Moreover, for \(n>1\) we have \(\sum_{d\mid n}\mu(d)=0\), which characterizes \(\mu\) among arithmetic functions.

For example, for \(n=30=2\cdot 3\cdot 5\) we get \(\mu(30)=(-1)^3=-1\), while \(\mu(12)=0\) because \(12=2^2\cdot 3\) has a repeated prime factor.

A typical mistake is to think that \(\mu(n)\) depends only on the number of prime factors. If \(n\) has a repeated factor, such as \(\mu(8)\), the value is \(0\), not \(-1\).

Think about it

What is \(\mu(18)\)?

0, because \(18=2\cdot 3^2\) has a repeated prime factor.

Exercises 10

  1. What is \(\mu(1)\)?
  2. What is \(\mu(7)\)?
  3. Which of these numbers has \(\mu(n)\neq 0\)?
  4. If \(n\) is a product of 4 distinct primes, what is \(\mu(n)\)?
  5. How many positive divisors of 12 satisfy \(\mu(d)\neq 0\)?
  6. You know that \(g(n)=\sum_{d\mid n} f(d)\) for every \(n\), with \(g(1)=2\), \(g(2)=5\), \(g(3)=4\) and \(g(6)=9\). What is \(f(6)\)?
  7. What is \(\mu(10)\cdot\mu(3)\)?
  8. Which of these statements about \(\mu\) is false?
  9. If \(p\) is prime and \(k\ge 2\), what is \(\mu(p^k)\)?
  10. Compute \(\sum_{d\mid 30}\mu(d)\).

References

  • David M. Burton, Elementary Number Theory, 7th edition, McGraw-Hill, 2011. cap. 6
  • Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley, 1991. cap. 4