An essay on the distribution of values of the delta-function and related topics

Abstract

The function of defined to be the sum of all positive integer divisors of n. This dissertation is a survey of major work done during the last 30 years on problems connected with numbers satisfying. For real values of X these numbers are called -abundant. When, the numbers are simply called abundant. When is any integer, but the equality sign only holds, the numbers satisfying the equation are called multi-perfect, or, for the special case, perfect. There is also some discussion on pairs of amicable numbers, for which. The first chapter contains those standard results of number theory which are essential for the development of this dissertation. The reader is advised to proceed direct to Chapter 11 and to refer to these results as they arise in the text. Chapter 11 is an introductory chapter, in which the historical background of these numbers is presented. It includes theorems discussing the general behaviour of the function, for example its average value, and true maximum order, and a deeper theorem finding the number of distinct numbers , for all integers m not exceeding. The third chapter contains a very full discussion of abundant and x -abundant numbers. Conditions for consecutive abundant numbers are found, the sequences of abundant numbers and -abundant numbers are shown to possess density, and bounds for the numbers of primitive abundant, and primitive X-abundant numbers not exceeding n are determined. Chapter IV concerns perfect and multi-perfect numbers. The density of the sequences of both these types of numbers is shown to be zero, and upper bounds for the numbers of both perfect and multi-perfect numbers not exceeding n are found. The final chapter contains a brief discussion of amicable numbers. Considerably less is known about these, and the chapter contains only one major theorem: namely that the density of the sequence of amitable numbers is zero. The theorems incorporated into this dissertation are mainly those of P.Erdos, whose work in this particular field has been outstanding.