The Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory.
are two arithmetic functions (i.e. functions from the positive integers
to the complex numbers
), one defines a new arithmetic function f
, the Dirichlet convolution
where the sum extends over all positive divisors d
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.
- To every multiplicative f there exists a multiplicative g such that f * g = ε.
With addition and Dirichlet convolution, the set of arithmetic functions forms a commutative ring
with multiplicative identity ε, the Dirichlet ring
. The units of this ring are the arithmetical functions 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 L-series by
for those complex
for which the series converges (if there are any). The multiplication of L-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 L-series as a Fourier transform