Statistics of the Mulitiplicative Groups

For every positive integer n, the quotient ring Z/nZ is the natural ring whose additive group is cyclic. The "multiplicative group modulo n" is the group of invertible elements of this ring, with the multiplication operation. As it turns out, many quantities of interest to number theorists can be interpreted as "statistics" of these multiplicative groups. For example, the cardinality of the multiplicative group modulo n is simply the Euler phi function of n; also, the number of terms in the invariant factor composition of this group is closely related to the number of primes dividing n. Many of these statistics have known distributions when the integer n is chosen at random (the Euler phi function has a singular cumulative distribution, while the Erdös–Kac theorem tells us that the number of prime divisors follows an asymptotically normal distribution). Therefore this family of groups provides a convenient excuse for examining several famous number theory results and open problems. We shall describe how we know, given the factorization of n, the exact structure of the multiplicative group modulo n, and go on to outline the connections to these classical statistical problems in multiplicative number theory.