The Möbius inversion or Möbius inversion formula goes back to August Ferdinand Möbius and allows a number-theoretic function to be reconstructed from its summation function .
A number theoretic function is given
![{\ displaystyle f \ colon \ mathbb {N} \ to \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/595c8b528a92771afe091895547d1780439d5d16)
and their summation function
![{\ displaystyle F \ colon \ mathbb {N} \ to \ mathbb {C}, \ quad F (n) = \ sum _ {d \ mid n} f (d).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95d95dd2bee9d26d68ea6051d168405ba1c26c84)
Then for every natural number n holds
![f (n) = \ sum _ {d \ mid n} \ mu (d) F \ left ({\ frac {n} {d}} \ right) = \ sum _ {d \ mid n} \ mu \ left ({\ frac {n} {d}} \ right) F (d),](https://wikimedia.org/api/rest_v1/media/math/render/svg/b631e2939e4e347fa9aed19b8e9212d7703d7d59)
where denotes the Möbius function .
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)