Sequences

by H. Halberstam, K.F. Roth

Overview

THIS volume is concerned with a substantial branch of number theory of which no connected account appears to exist; we describe the general nature of the constituent topics in the introduction. Although some excellent surveys dealing with limited aspects of the subject under con­ sideration have been published, the literature as a whole is far from easy to study. This is due in part to the extent of the literature; it is necessary to thread one's way through a maze of results, a complicated structure of inter-relationships, and many conflicting notations. In addition, however, not all the original papers are free from obscurities, and consequently some of these papers are difficult (a few even exceed­ ingly difficult) to master. We try to give a readable and coherent account of the subject, con­ taining a cross-section of the more interesting results. We felt that it would have been neither practicable nor desirable to attempt a compre­ hensive account; we treat each aspect of the subject from some special point of view, and select results accordingly. Needless to say, this approach entails the omission of many interesting and important results (quite apart from defects in the selection due to errors of judgement on our part). Those results selected for inclusion are, however, proved in complete detail and without the assumption of any prior knowledge on the part of the reader.

Product Details

ISBN-13:	9781461382294
Publisher:	Springer New York
Publication date:	10/14/2011
Edition description:	Softcover reprint of the original 1st ed. 1983
Pages:	293
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

I. Addition of Sequences: Study of Density Relationships.- § 1. Introduction and notation.- § 2. Schnirelmann density and Schnirelmann’s theorems. Besicovitch’s theorem.- § 3. Essential components and complementary sequences.- § 4. The theorems of Mann, Dyson, and van der Corput.- § 5. Bases and non-basic essential components.- § 6. Asymptotic analogues and p-adic analogues.- § 7. Kneser’s theorem.- § 8. Kneser’s theorem (continued): the ?-transformations.- § 9. Kneser’s theorem (continued): proof of Theorem 19—sequence functions associated with the derivations of a system.- § 10. Kneser’s theorem (continued): proofs of Theorems 16? and 17?.- § 11. Hanani’s conjecture.- II. Addition of Sequences: Study of Representation Functions by Number Theoretic Methods.- § 1. Introduction.- § 2. Auxiliary results from the theory of finite fields.- § 3. Sidon’s problems.- § 4. The Erdös—Fuchs theorem.- III. Addition of Sequences: Study of Representation Functions by Probability Methods.- § 1. Introduction.- § 2. Principal results.- § 3. Finite probability spaces: informal discussion.- § 4. Measure theory: basic definitions.- § 5. Measure theory: measures on product spaces.- § 6. Measure theory: simple functions.- § 7. Probability theory: basic definitions and terminology.- § 8. Auxiliary lemmas.- § 9. Probability theory: some fundamental theorems.- § 10. Probability measures on the space of (positive) integer sequences.- § 11. Preparation for the proofs of Theorems 1–4.- § 12. Proof of Theorem 1.- § 13. Proof of Theorem 2.- § 14. Proof of Theorem 3.- § 15. Quasi-independence of the variables rn.- §16. Proof of Theorem 4—sequences of pseudo-squares.- IV. Sieve Methods.- § 1. Introduction.- § 2. Notation and preliminaries.- § 3. The number of natural numbers not exceeding x not divisible by any prime less than y.- § 4. The generalized sieve problem.- § 5. The Viggo Brun method.- § 6. Selberg’s upper-bound method: informal discussion.- §7. Selberg’s upper-bound method.- § 8. Selberg’s lower-bound method.- § 9. Selberg’s lower-bound method: further discussion.- § 10. The ‘large’ sieves of Linnik and Rényi.- V. Primitive Sequences and Sets of Multiples.- § 1. Introduction.- § 2. Density.- § 3. An inequality concerning densities of unions of congruence classes.- § 4. Primitive sequences.- § 5. The set of multiples of a sequence: applications including the proofs of Theorems 4 and 5.- § 6. A necessary and sufficient condition for the set of multiples of a given sequence to possess asymptotic density.- § 7. The set of multiples of a special sequence.- § 8. Proof of Theorem 15.- § 2. The distribution of prime numbers.- § 3. Mean values of certain arithmetic functions.- § 4. Miscellanea from elementary number theory.- References.- Postscript.- Author Index.