Möbius inversion

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

${\displaystyle f\colon \mathbb {N} \to \mathbb {C} }$

and their summation function

${\displaystyle F\colon \mathbb {N} \to \mathbb {C} ,\quad F(n)=\sum _{d\mid n}f(d).}$

Then for every natural number n holds

${\displaystyle f(n)=\sum _{d\mid n}\mu (d)F\left({\frac {n}{d}}\right)=\sum _{d\mid n}\mu \left({\frac {n}{d}}\right)F(d),}$

where denotes the Möbius function . ${\displaystyle \mu }$