アルティン・ハッセの指数関数
where "H" runs over open subgroups of "G" with finite index (there are finitely many of each index since "G" is topologically finitely generated) and "a" is the number of continuous homomorphisms from "G" to "S". Two special cases are worth noting. (1) If "G" is the "p"-adic integers, it has exactly one open subgroup of each "p"-power index and a continuous homomorphism from "G" to "S" is essentially the same thing as choosing an element of "p"-power order in "S", so we have recovered the above combinatorial interpretation of the Taylor coefficients in the Artin–Hasse exponential series. (2) If "G" is a finite group then the sum in the exponential is a finite sum running over all subgroups of "G", and continuous homomorphisms from "G" to "S" are simply homomorphisms from "G" to "S". The result in this case is due to Wohlfahrt (1977). The special case when "G" is a finite cyclic group is due to Chowla, Herstein, and Scott (1952), and takes the form