Editing Dirichlet convolution (section)

==Dirichlet series==
If ''f'' is an arithmetic function, the [[Dirichlet series]] [[generating function]] is defined by

:<math>
DG(f;s) = \sum_{n=1}^\infty \frac{f(n)}{n^s}
</math>

for those [[complex number|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:

:<math>
DG(f;s) DG(g;s) = DG(f*g;s)\,
</math>

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]].