Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory. This was developed by Johann Peter Gustav Lejeune Dirichlet, a German mathematician.
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.
Some 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.
- For each f for which f(1) ≠ 0 there exists a g such that f * g = ε. g is called the Dirchlet inverse of f.
- In particular, every multiplicative f has a Dirichlet inverse g that is also multiplicative.
With addition and Dirichlet convolution, the set of arithmetic functions forms a commutative ring with multiplicative identity ε, the Dirichlet ring (note that it is not a field because some arithmetic functions do not have Dirichlet inverses). The units of this ring are the arithmetical functions f with f(1) ≠ 0.
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 Dirichlet series generating function by
for those complex arguments s for which the series converges (if there are any). The multiplication of Dirichlet series is compatible with Dirichlet convolution in the following sense:
for all s for which the left hand side exists. This is akin to the convolution theorem if one thinks of Dirichlet series as a Fourier transform.