Dirichlet Series - Combinatorial Importance

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 a_n 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