Studies in multiplicative number theory - Shiu, Peter Man-Kit (1980)
This thesis gives some order estimates and asymptotic formulae associated with general classes of non-negative multiplicative functions as well as some particular multiplicative functions such as the divisor functions dk(n). In Chapter One we give a lower estimate for the number of distinct values assumed by the divisor function d(n) in 1 <n <x .We also identify the smallest positive integer which is a product of triangular numbers and not equal to d3(n) for 1 <n <x . In Chapter Two we show that if f(n) satisfies some conditions and if M=max {f(2a)}1/a, if a> or = 1 then the maximum value of f(n) in 1<n< x is about log x / Mloglog x. We also show that a function which has a finite mean value cannot be large too often. In Chapter Three we give an upper estimate to the average value of f(n) as n runs through a short interval in an arithmetic progression with a large modulus . As an application of our general theorem we show, for example, that if f(n) is the characteristic function of the set of integers which are the sum of two squares, then as x -> infinity. We call a positive integer n a k-full integer if pk divides n whenever p is a prime divisor of n, and in Chapter Four we give an asymptotic formula for the number of k-full integers not exceeding x. In Chapter Five we give an asymptotic formula for the number of 2-full integers in an interval. We also study the problem of the distribution of the perfect squares among the sequence of 2-full integers. The materials in the first three chapters have been accepted for publications and will appear [31], [22], [33] and [32].