Generalized Convolutions

Ifis a real or complex valued function defined on the positive real axissuch that forthenis called the generalized convolution ofwithwhereis an arithmetical function.

Iffor all nonintegralthe restriction ofto the integers is an arithmetical function andso the operationcan be regarded as a generalisation of the Dirichlet convolution.

Unlike the Dirichlet convolution the operationis neither commutative not associative, but it is associative in the following sense:

Theorem 1

whereandare arithmetical functions.

Proof

Hence

The identity function for Dirichlet convolution is also the identity function for generalised convolution:

We can use this to prove the following inversion formula.

Theorem 2

If an arithmetical function has a Dirichlet inversethen the equationimpliesand vice versa.

Proof: IfthenProof of the converse is similar.

Ifis completely multiplicative thenthen impliesand vice versa.