Accession Number : AD0614834

Title :   ASYMPTOTIC DISTRIBUTION OF GENERALIZED INTEGERS AND PRIME NUMBERS,

Corporate Author : STANFORD UNIV CALIF DEPT OF MATHEMATICS

Personal Author(s) : Diamond,Harold G.

Report Date : 29 APR 1965

Pagination or Media Count : 62

Abstract : The paper is divided into four chapters, of which the first two are concerned with a wide class of measures and general properties of these measures. The last two chapters deal with a few measures of prime number theoretic interest, and for these measures some special results are proved. In the first chapter 'C' and 'R' classes of measures are introduced. Convolution and certain other operations are defined and their elementary properties explored. A topology is induced in 'C' by total variation semi norms, and 'C' is shown to be a complete topological group under addition. The second chapter begins with some general results about convergence and continuity of power series in measures. C sub 1 denotes the elements of 'C' with point mass at 1, and all measures in C sub 1 is shown to be invertible in a convolution sense. The main result of the chapter is the representation of C sub 1 by 'C' under the exponential mapping. This mapping is shown to be a homeomorphism and an isomorphism of topological groups. As a corollary to this theorem the existence of nth roots of measures in C sub 1 is infered. G-primes and integers and their related measures are defined in the third chapter. Using the results of Chapter 2, the counting measure of gintegers and Lebesgue measure is shown to be given as exponentials. The prime number theorem for g-primes is the main result of the fourth chapter.

Descriptors :   (*PRIME NUMBERS, STATISTICAL DISTRIBUTIONS), (*NUMBER THEORY, PRIME NUMBERS), REAL NUMBERS, SET THEORY, TOPOLOGY, STATISTICAL FUNCTIONS, THEOREMS

Distribution Statement : APPROVED FOR PUBLIC RELEASE