Dirichlet convolution

The connection between and can be written in an elegant way using a Dirichlet convolution:

It is called the unit function because it is the identity element for Dirichlet convolution.

See Dirichlet convolution, below.

This function c is called the Dirichlet convolution of a and b, and is denoted by .

The multiplication of Dirichlet series is compatible with Dirichlet convolution in the following sense:

This is also a consequence of the fact that we can write as a Dirichlet convolution of where is the characteristic function of the squares.

In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory.

This and many of the following series may be obtained by applying Möbius inversion and Dirichlet convolution to known series.

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.

Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the product of two power series:

Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions and , let be their Dirichlet convolution.

The Dirichlet convolution of two multiplicative functions is again multiplicative, and every multiplicative function has a Dirichlet inverse that is also multiplicative.

While the Dirichlet convolution of two multiplicative functions is multiplicative, the Dirichlet convolution of two completely multiplicative functions need not be completely multiplicative.