Combinatorial Importance
Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian products.
Suppose that A is a set with a function w: A → N assigning a weight to each of the elements of A, and suppose additionally that the fibre over any natural number under that weight is a finite set. (We call such an arrangement (A,w) a weighted set.) Suppose additionally that an is the number of elements of A with weight n. Then we define the formal Dirichlet generating series for A with respect to w as follows:
Note that if A and B are disjoint subsets of some weighted set (U, w), then the Dirichlet series for their (disjoint) union is equal to the sum of their Dirichlet series:
Moreover, and perhaps a bit more interestingly, if (A, u) and (B, v) are two weighted sets, and we define a weight function w: A × B → N by
for all a in A and b in B, then we have the following decomposition for the Dirichlet series of the Cartesian product:
This follows ultimately from the simple fact that
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