Sequences - H. Halberstam

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Sommaire:
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 Erd?s-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 R?nyi.- 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.