Sequences, Vol 1

Includes index

Includes bibliographical references

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 § 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 § 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 § 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

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.