Multiplicative Function - Convolution

Convolution

If f and g are two multiplicative functions, one defines a new multiplicative function f * g, the Dirichlet convolution of f and g, by

where the sum extends over all positive divisors d of n. With this operation, the set of all multiplicative functions turns into an abelian group; the identity element is .

Relations among the multiplicative functions discussed above include:

• * 1 = (the Möbius inversion formula)
• ( * Id_k) * Id_k = (generalized Möbius inversion)
• * 1 = Id
• d = 1 * 1
• = Id * 1 = * d
• _k = Id_k * 1
• Id = * 1 = *
• Id_k = _k *

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