The Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory.
See also:
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, bySome general properties of this operation include:
- If both f and g are multiplicative, then so is f * g. (Note however that the convolution of two completely multiplicative functions need not be completely multiplicative.)
- f * g = g * f (commutativity)
- (f * g) * h = f * (g * h) (associativity)
- f * (g + h) = f * g + f * h (distributivity)
- f * ε = ε * f = f, where ε is the function defined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1.
- To every multiplicative f there exists a multiplicative g such that f * g = ε.
Furthermore, the multiplicative functions with convolution form an abelian group with identity element ε. The article on multiplicative functions lists several convolution relations among important multiplicative functions.
If f is an arithmetic function, one defines its L-series by