The mobius function and the mobius inversion formula

NUMBER THEORY
The mobius function and the mobius inversion formula

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) 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) =  

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

14. Thank You!