Dirichlet convolution

In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.

If f and g are two arithmetic functions (i.e. functions from the positive integers to the complex numbers), one defines a new arithmetic function f ∗ g, the Dirichlet convolution of f and g, by

where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs (a, b) of positive integers whose product is n.

The set of arithmetic functions forms a commutative ring, the Dirichlet ring, under pointwise addition (i.e. f + g is defined by (f + g)(n) = f(n) + g(n)) and Dirichlet convolution. The multiplicative identity is the unit function ε defined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1. The units (i.e. invertible elements) of this ring are the arithmetic functions f with f(1) ≠ 0.

Specifically, Dirichlet convolution isassociative,

distributes over addition

is commutative,

and has an identity element,

Furthermore, for each f for which f(1) ≠ 0 there exists a g such that f ∗ g = ε, called the Dirichlet inverse of f.

...
Wikipedia