Möbius transform

The Möbius transform should not be confused with Möbius transformations.

In mathematics, the Möbius transform Tf of a function f defined on the positive integers is defined by

<math>(Tf)(n)=\sum_{d\mid n} f(d)\mu(n/d)=\sum_{d\mid n} f(n/d)\mu(d)<math>

where μ is the classic Möbius function. In more common usage, the function Tf is called the Möbius inverse of f.

(The notation d | n means d is a divisor of n.)

This function is named in honor of August Ferdinand Möbius.

It takes multiplicative functions to multiplicative functions. On Dirichlet series generating functions it corresponds to division by the Riemann zeta function.

See also: Möbius inversion formula.