Distribution of Random Variables on the Symmetric Group

Abstract

The well known Erdos-Turan law states that the logarithm of an order of a random permutation is asymptotically normally distributed. The aim of this work is to estimate convergence rate in this theorem and also to prove analogous result for distribution of the logarithm of an order of a random permutation on a certain class of subsets of the symmetric group. We also study the asymptotic behavior of the mean values of multiplicative functions on the symmetric group and the results we obtain are of independent interest besides their application to the investigation of the remainder term in the Erdos-Turan law. We also study a related problem of distribution of the degree of a splitting field of a random polynomial and obtain sharp estimates for its convergence rate to normal law. In research we apply both probabilistic and analytic methods. Some analytic methods used here have their origins in the probabilistic number theory, and some have their roots in the theory of summation of divergent series. One of the approaches we use is to apply Tauberian type estimates for Voronoi summability of divergent series to analyze the generating functions of the mean values of multiplicative functions.