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
which, in turn, is the basis for the Möbius inversion formula relating sums over divisors:
Special cases of Möbius inversion pairs are:
Other types of Möbius inversion formulas include:
For a general theory of Möbius inversion with applications to combinatorial theory see Rota (1964).