of the formula for the value of the geometric series. From this we deduce that can be expressed in terms of the Riemann zeta function as For more complicated examples, the computations become difficult, and in general one cannot expect a closed expression for . The local factor can always be expressed as a definable -adic integral. Applying a result of MacIntyre on the model theory of -adic integers, one deduces again that is a rational function in . Moreover, M. du Sautoy and F. Grunewald showed that the integral can be approximated by Artin L-functions. Using the fact that Artin L-functions are holomorphic in a neighbourhood of the line , they showed that for any torsionfree nilpotent group, the function is meromorphic in the domain where is the abscissa of convergence of , and is some positive number, and holomorphic in some neighbourhood of . Using a Tauberian theorem this implies for some real number and a non-negative integer . Congruence subgroups Subgroup growth and coset representations Let be a group, a subgroup of index . Then acts on the set of left cosets of in by left shift: In this way, induces a homomorphism of into the symmetric group on . acts transitively on , and vice versa, given a transitive action of on the stabilizer of the point 1 is a subgroup of index in . Since the set can be permuted in ways, we find that is equal to the number of transitive -actions divided by . Among all -actions, we can distinguish transitive actions by a sifting argument, to arrive at the following formula where denotes the number of homomorphisms In several instances the function is easier to be approached then , and, if grows sufficiently large, the sum is of negligible order of magnitude, hence, one obtains an asymptotic formula for . As an example, let be the free group on two generators. Then every map of the generators of extends to a homomorphism that is From this we deduce For more complicated examples, the estimation of involves the representation theory and statistical properties of symmetric groups. References Infinite group theory Zeta and L-functions