Anatomy of smooth integers
The object of the first chapter of this thesis is to review the materials and tools in analytic number theory which are used in following chapters. We also give a survey on the development concerning the number of y−smooth integers, which are integers free of prime factors greater than y. In the second chapter, we shall give a brief history about a class of arithmetical functions on a probability space and we discuss on some well-known problems in probabilistic number theory. We present two results in analytic and probabilistic number theory. The Erdos multiplication table problem asks what is the number of distinct integers appearing in the N × N multiplication table. The order of magnitude of this quantity was determined by Kevin Ford (2008). In chapter 3 of this thesis, we study the number of y−smooth entries of the N × N multiplication. More concretely, we focus on the change of behaviour of the function A(x,y) in different ranges of y, where A(x,y) is a function that counts the number of distinct y−smooth integers less than x which can be represented as the product of two y−smooth integers less than p x. In Chapter 4, we prove an Erdos-Kac type of theorem for the set of y−smooth integers. If !(n) is the number of distinct prime factors of n, we prove that the distribution of !(n) is Gaussian for a certain range of y using method of moments.
Analytic number theory Probabilistic number theory Method of moments y−smooth integers Erdos multiplication table problem Erdos-Kac theorem