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.
Definition.
If formula_1 are two arithmetic functions from the positive integers to the complex numbers, the "Dirichlet convolution" is a new arithmetic function defined by:
where the sum extends over all positive divisors "d" of "n", or equivalently over all distinct pairs of positive integers whose product is "n".
This product occurs naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients:
Properties.
The set of arithmetic functions forms a commutative ring, the , under pointwise addition, where is defined by , and Dirichlet convolution. The multiplicative identity is the unit function "ε" defined by if and if . The units (invertible elements) of this ring are the arithmetic functions "f" with .
Specifically, Dirichlet convolution is associative,
distributes over addition
is commutative,
and has an identity element,
Furthermore, for each formula_9 having formula_10, there exists an arithmetic function formula_11 with formula_12, called the of formula_9.
The Dirichlet convolution of two multiplicative functions is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse which is also multiplicative. In other words, multiplicative functions form a subgroup of the group of invertible elements of the Dirichlet ring. Beware however that the sum of two multiplicative functions is not multiplicative (since formula_14), so the subset of multiplicative functions is not a subring of the Dirichlet ring. The article on multiplicative functions lists several convolution relations among important multiplicative functions.
Another operation on arithmetic functions is pointwise multiplication: is defined by . Given a completely multiplicative function formula_15, pointwise multiplication by formula_15 distributes over Dirichlet convolution: formula_17. The convolution of two completely multiplicative functions is multiplicative, but not necessarily completely multiplicative.
Examples.
In these formulas, we use the following arithmetical functions:
The following relations hold:
This last identity shows that the prime counting function is given by the summatory function 
where formula_61 is the Mertens function and formula_62 is the distinct prime factor counting function from above. This expansion follows from the identity for the sums over Dirichlet convolutions given on the divisor sum identities page (a standard trick for these sums).
Dirichlet inverse.
Examples.
Given an arithmetic function formula_9 its Dirichlet inverse formula_64 may be calculated recursively: the value of formula_65 is in terms of formula_66 for formula_67.
For formula_68:
For formula_73:
For formula_76:
For formula_79:
and in general for formula_82,
Properties.
The following properties of the Dirichlet inverse hold:
Other formulas.
An exact, non-recursive formula for the Dirichlet inverse of any arithmetic function "f" is given in Divisor sum identities. A more partition theoretic expression for the Dirichlet inverse of "f" is given by 
The following formula provides a compact way of expressing the Dirichlet inverse of an invertible arithmetic function "f" :
formula_90
where the expression formula_91 stands for the arithmetic function formula_92 convoluted with itself "k" times. Notice that, for a fixed positive integer formula_22, if formula_94 then formula_95 , this is because formula_96 and every way of expressing "n" as a product of "k" positive integers must include a 1, so the series on the right hand side converges for every fixed positive integer "n." 
Dirichlet series.
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 both series of the left hand side converge, one of them at least converging 
absolutely (note that simple convergence of both series of the left hand side DOES NOT imply convergence of the right hand side!). This is akin to the convolution theorem if one thinks of Dirichlet series as a Fourier transform.
Related concepts.
The restriction of the divisors in the convolution to unitary, bi-unitary or infinitary divisors defines similar commutative operations which share many features with the Dirichlet convolution (existence of a Möbius inversion, persistence of multiplicativity, definitions of totients, Euler-type product formulas over associated primes, etc.).
Dirichlet convolution is the convolution of the incidence algebra for the positive integers ordered by divisibility.