structural:reformatting
Dirichlet convolution

Given an arithmetic function a (n), let F (s), for complex s, be the function defined by the corresponding Dirichlet series (where it converges):

F (s) is called a generating function of a (n). The simplest such series, corresponding to the constant function a (n) = 1 for all n, is ζ (s) the Riemann zeta function.

The generating function of the Möbius function is the inverse of the zeta function:

Consider two arithmetic functions a and b and their respective generating functions F (s) and F (s). The product F (s) F (s) can be computed as follows:

It is a straightforward exercise to show that if c (n) is defined by

then

This function c is called the Dirichlet convolution of a and b, and is denoted by a ∗ b {\displaystyle a*b}.

A particularly important case is convolution with the constant function a (n) = 1 for all n, corresponding to multiplying the generating function by the zeta function:

Multiplying by the inverse of the zeta function gives the Möbius inversion formula:

If f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative.