Limit theorems for B–free integers and the Moebius function

Abstract

This thesis presents several limit theorems linked by the topic of pseudorandomness of the Moebius function.

The Moebius function is supported on the sequence of square–free integers. The distribution of square–free integers in short intervals was studied by Hall in a series of papers starting 1982. In particular, he provided the asymptotic order for the mean square of the number of square–free integers in a short interval and proved upper bounds for higher moments of the same variable. The first part of this thesis contains a generalization of Hall’s result on the mean square to a class of B–free integers under certain assumptions on the underlying set B. Additionally, we provide a weaker analogue of these results for the case of number fields.

The second part of this thesis is concerned with the limiting distribution of two random variables with respect to a special class of logarithmic measures appearing in a study of the Mertens function. An adaptation of Erdos–Kac theorem to these measures is presented; more precisely, it is shown that the limiting distribution of the number of different prime factors of an integer with respect to our signed measures has a positive and a negative Gaussian parts. Further, the distribution of smooth integers in the same framework is studied. The limiting distribution in this case is also explicitly given and turns out to be intricately connected to the so-called generalized Dickman–de Bruijn distribution.

Overall, the results presented in this thesis lead to a better understanding of statistics of several arithmetical sequences.