Editing Dirichlet convolution (section)

==Definition==
If <math>f , g : \mathbb{N}\to\mathbb{C}</math> are two [[Arithmetic function|arithmetic functions]], their Dirichlet convolution <math>f*g</math> is a new arithmetic function defined by:

:<math>
(f*g)(n) \ =\ \sum_{d\,\mid \,n} f(d)\,g\!\left(\frac{n}{d}\right) \ =\  \sum_{ab\,=\,n}\!f(a)\,g(b),
</math>

where the sum extends over all positive [[divisor]]s <math>d</math> of <math>n</math>, or equivalently over all distinct pairs <math>(a,b)</math> of positive integers whose product is <math>n</math>.

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:
:<math>\left(\sum_{n\geq 1}\frac{f(n)}{n^s}\right)
\left(\sum_{n\geq 1}\frac{g(n)}{n^s}\right)
\ = \ 
\left(\sum_{n\geq 1}\frac{(f*g)(n)}{n^s}\right).
</math>