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 19sequence 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 ErdösFuchs 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 14. - 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 4sequences of pseudo-squares. - IV. Sieve Methods. - 1. Introduction. - 2. Notation andpreliminaries. - 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. Language: English