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denotes the Dirichlet convolution, and is the constant function . The second formula is then written as

:

Many specific examples are given in the article on multiplicative functions.

The theorem follows because is (commutative and) associative, and , where is the identity function for the Dirichlet convolution, taking values , for all . Thus
:.Replacing by , we obtain the product version of the Möbius inversion formula:

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