# Möbius inversion for arithmetic functions

The original form of Möbius inversion developed by A. Möbius for arithmetic functions.

Möbius considered also inversion formulas for finite sums running over the divisors of a natural number $n$: $$F(n) = \sum_{d | n} f(d) \ ,\ \ \ f(n) = \sum_{d | n} \mu(d) F(n/d) \ .$$ The correspondence $f \mapsto F$ is the Möbius transform, and $F \mapsto f$ the inverse Möbius transform.

Another inversion formula: If $P(n)$ is a totally multiplicative function for which $P(1) = 1$, and $f(x)$ is a function defined for all real $x > 0$, then $$g(x) = \sum_{n \le x} P(n) f(x/n)$$ implies $$f(x) = \sum_{n \le x} \mu(n) P(n) g(x/n) \ .$$

All these (and many other) inversion formulas follow from the basic property of the Möbius function that it is the inverse of the unit arithmetic function $E(n) \equiv 1$ under Dirichlet convolution, cf. (the editorial comments to) Möbius function and Multiplicative arithmetic function.

How to Cite This Entry:
Richard Pinch/sandbox-5. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-5&oldid=51489