The objective of this thesis is to estimate the functions [mathematical formula removed] for n and m integers, and [mathematical formula removed] for q and r primes. In Chapter One we give a series of lemmas relating to the ensuing chapters. In Chapter Two we deal with the function F(x,y,z) for a=b=c=0, and in Chapter Three with P(x,y,z) for a=b=c=0. In Chapters Four and Five the major theorems of the thesis are presented.