The Möbius inversion formula in number theory states that if g(n) and f(n) are arithmetic functions satisfying
- <math>g(n)=\sum_{d\mid n}f(d)\quad\mbox{for every integer }n\ge 1</math>
then
- <math>f(n)=\sum_{d\mid n}g(d)\mu(n/d)\quad\mbox{for every integer }n\ge 1</math>
where μ is the
Möbius function and the sums extend over all positive
divisors d of
n.
The formula is also correct if f and g are functions from the positive integers into some abelian group.
In the language of convolutions (see multiplicative function), the inversion formula can also be expressed as
- μ * 1 = ε.
An equivalent formulation of the inversion formula more useful in
combinatorics is as follows: suppose
F(
x) and
G(
x) are
complex-valued
functions defined on the
interval [1,∞) such that
- <math>G(x) = \sum_{1 \le n \le x}F(x/n)\quad\mbox{ for all }x\ge 1</math>
then
- <math>F(x) = \sum_{1 \le n \le x}\mu(n)G(x/n)\quad\mbox{ for all }x\ge 1.</math>
Here the sums extend over all positive integers n which are less than or equal to x.