Sequences

I. Addition of Sequences: Study of Density Relationships.- x 1. Introduction and notation.- x 2. Schnirelmann density and Schnirelmann's theorems. Besicovitch's theorem.- x 3. Essential components and complementary sequences.- x 4. The theorems of Mann, Dyson, and van der Corput.- x 5. Bases and non-basic essential components.- x 6. Asymptotic analogues and p-adic analogues.- x 7. Kneser's theorem.- x 8. Kneser's theorem (continued): the ?-transformations.- x 9. Kneser's theorem (continued): proof of Theorem 19-sequence functions associated with the derivations of a system.- x 10. Kneser's theorem (continued): proofs of Theorems 16? and 17?.- x 11. Hanani's conjecture.- II. Addition of Sequences: Study of Representation Functions by Number Theoretic Methods.- x 1. Introduction.- x 2. Auxiliary results from the theory of finite fields.- x 3. Sidon's problems.- x 4. The Erdos-Fuchs theorem.- III. Addition of Sequences: Study of Representation Functions by Probability Methods.- x 1. Introduction.- x 2. Principal results.- x 3. Finite probability spaces: informal discussion.- x 4. Measure theory: basic definitions.- x 5. Measure theory: measures on product spaces.- x 6. Measure theory: simple functions.- x 7. Probability theory: basic definitions and terminology.- x 8. Auxiliary lemmas.- x 9. Probability theory: some fundamental theorems.- x 10. Probability measures on the space of (positive) integer sequences.- x 11. Preparation for the proofs of Theorems 1-4.- x 12. Proof of Theorem 1.- x 13. Proof of Theorem 2.- x 14. Proof of Theorem 3.- x 15. Quasi-independence of the variables rn.- x16. Proof of Theorem 4-sequences of pseudo-squares.- IV. Sieve Methods.- x 1. Introduction.- x 2. Notation and preliminaries.- x 3. The number of natural numbers not exceeding x not divisible by any prime less than y.- x 4. The generalized sieve problem.- x 5. The Viggo Brun method.- x 6. Selberg's upper-bound method: informal discussion.- x7. Selberg's upper-bound method.- x 8. Selberg's lower-bound method.- x 9. Selberg's lower-bound method: further discussion.- x 10. The 'large' sieves of Linnik and Renyi.- V. Primitive Sequences and Sets of Multiples.- x 1. Introduction.- x 2. Density.- x 3. An inequality concerning densities of unions of congruence classes.- x 4. Primitive sequences.- x 5. The set of multiples of a sequence: applications including the proofs of Theorems 4 and 5.- x 6. A necessary and sufficient condition for the set of multiples of a given sequence to possess asymptotic density.- x 7. The set of multiples of a special sequence.- x 8. Proof of Theorem 15.- x 2. The distribution of prime numbers.- x 3. Mean values of certain arithmetic functions.- x 4. Miscellanea from elementary number theory.- References.- Postscript.- Author Index.