The Möbius transform should not be confused with Möbius transformations.

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

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


where μ is the classic Möbius function

Möbius function

The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...

. In more common usage, the function Tf is called the Möbius inverse

Möbius inversion formula

In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. Other Möbius inversion formulas are obtained when different local finite partially ordered sets replace the classic case of the natural numbers ordered by...

 of f.

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

This function is named in honor of August Ferdinand Möbius

August Ferdinand Möbius

August Ferdinand Möbius was a German mathematician and theoretical astronomer.He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict...

.

The transform takes multiplicative function

Multiplicative function

In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...

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

Series relations

Let


so that


be its transform. The transforms are related by means of series: the Lambert series


and the Dirichlet series:


where is the Riemann zeta function.