# §27.5 Inversion Formulas

If a Dirichlet series generates , and generates , then the product generates

called the Dirichlet product (or convolution) of and . The set of all number-theoretic functions with forms an abelian group under Dirichlet multiplication, with the function in (27.2.5) as identity element; see Apostol (1976, p. 129). The multiplicative functions are a subgroup of this group. Generating functions yield many relations connecting number-theoretic functions. For example, the equation is equivalent to the identity

27.5.2

which, in turn, is the basis for the Möbius inversion formula relating sums over divisors:

27.5.3

Special cases of Möbius inversion pairs are:

Other types of Möbius inversion formulas include:

27.5.8

For a general theory of Möbius inversion with applications to combinatorial theory see Rota (1964).