This document discusses the Mobius function and the Mobius inversion formula. It defines the Mobius function μ(n) which investigates integers in terms of their prime decomposition. It then defines the Mobius inversion formula which determines the values of a function f at an integer in terms of its summatory function F. It proves several theorems about the Mobius function and Mobius inversion formula, including that the Mobius function is multiplicative and that the Mobius inversion formula can be used to invert summatory functions and retrieve the original function.

1. THE MOBIUS FUNCTION AND
THE MOBIUS INVERSION
FORMULA
MULTIPLICATIVE NUMBER THEORETIC FUNCTIONS

2. Definition of Mobius Function and the Mobius
Inversion Formula
• Mobius Function
• investigates integers in terms of their prime decomposition.
• Mobius Inversion Formula
• determines the values of the a function f at a given integer
in terms of its summatory function.

3. Definition
µ(n)
1 𝑖𝑓 𝑛 = 1; (−1)𝑡 𝑖𝑓 𝑛 = 𝑝1𝑝2. . .
𝑝𝑡 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑝𝑖 𝑎𝑟𝑒 𝑑𝑖𝑠𝑡𝑖𝑛𝑐𝑡 𝑝𝑟𝑖𝑚𝑒𝑠;
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
Note that if n is divisible by a power of a prime higher than one then µ(n) = 0. In connection with the
above definition, we have the following

4. Definition
An integer n is said to be square-free, if no square divides it, i.e. if
there does not exist an integer k such that k2 | n.
It is immediate (prove as exercise) that the prime-number
factorization of a square-free integer contains only distinct primes.

5. …Example
Notice that µ(1) = 1, µ(2) = −1, µ(3) = −1 and µ(4) = 0. We
now prove that µ(n) is a multiplicative function.

6. THE MOBIUS FUNCTION

7. Theorem
Theorem: The Mobius function µ(n) is multiplicative.
Proof. Let m and n be two relatively prime integers.
We have to prove that
µ(mn) = µ(m)µ(n).

8. THE MOBIUS INVERSION FORMULA

9. Theorem
Theorem: Let F(n) = ∑d|n µ(d), then F(n) satisfies
F(n) =
1 if n = 1;
0 if n > 1.

10. …Proof
Proof. For n = 1, we have F(1) = µ(1) = 1. Let us now find µ(pk)
for any integer k > 0. Notice that
F(pk) = µ(1) + µ(p) + ... + µ(pk) = 1 + (−1) + 0 + ... + 0 = 0

11. Theorem
Theorem: Suppose that f is an arithmetic function and suppose
that F is its summatory function, then for all positive integers n
we have

12. …Proof
Proof. We have

13. …Example
Example. A good example of a Mobius inversion formula would
be the inversion of σ(n) and τ(n). These two functions are the
summatory functions of f(n) = n and f(n) = 1 respectively. Thus
we get
and