アルティン・ハッセの指数関数
.
More generally, for any topologically finitely generated profinite group G there is an identity
exp  ( ∑ H ⊂ G x [ G : H ] / [ G : H ] ) = ∑ n ≥ 0 a G , n n ! x n , {\displaystyle \exp(\sum _{H\subset G}x^{[G:H]}/[G:H])=\sum _{n\geq 0}{\frac {a_{G,n}}{n!}}x^{n},}
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 aG,n is the number of continuous homomorphisms from G to Sn. Two special cases are worth noting.