Title | Sequences [electronic resource] / edited by H. Halberstam, K. F. Roth |
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Imprint | New York, NY : Springer New York, 1983 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8227-0 |
Descript | 293p. online resource |
I. Addition of Sequences: Study of Density Relationships -- ยง 1. Introduction and notation -- ยง 2. Schnirelmann density and Schnirelmannโs theorems. Besicovitchโs theorem -- ยง 3. Essential components and complementary sequences -- ยง 4. The theorems of Mann, Dyson, and van der Corput -- ยง 5. Bases and non-basic essential components -- ยง 6. Asymptotic analogues and p-adic analogues -- ยง 7. Kneserโs theorem -- ยง 8. Kneserโs theorem (continued): the ?-transformations -- ยง 9. Kneserโs theorem (continued): proof of Theorem 19โsequence functions associated with the derivations of a system -- ยง 10. Kneserโs theorem (continued): proofs of Theorems 16? and 17? -- ยง 11. Hananiโs conjecture -- II. Addition of Sequences: Study of Representation Functions by Number Theoretic Methods -- ยง 1. Introduction -- ยง 2. Auxiliary results from the theory of finite fields -- ยง 3. Sidonโs problems -- ยง 4. The ErdรถsโFuchs theorem -- III. Addition of Sequences: Study of Representation Functions by Probability Methods -- ยง 1. Introduction -- ยง 2. Principal results -- ยง 3. Finite probability spaces: informal discussion -- ยง 4. Measure theory: basic definitions -- ยง 5. Measure theory: measures on product spaces -- ยง 6. Measure theory: simple functions -- ยง 7. Probability theory: basic definitions and terminology -- ยง 8. Auxiliary lemmas -- ยง 9. Probability theory: some fundamental theorems -- ยง 10. Probability measures on the space of (positive) integer sequences -- ยง 11. Preparation for the proofs of Theorems 1โ4 -- ยง 12. Proof of Theorem 1 -- ยง 13. Proof of Theorem 2 -- ยง 14. Proof of Theorem 3 -- ยง 15. Quasi-independence of the variables rn -- ยง16. Proof of Theorem 4โsequences of pseudo-squares -- IV. Sieve Methods -- ยง 1. Introduction -- ยง 2. Notation and preliminaries -- ยง 3. The number of natural numbers not exceeding x not divisible by any prime less than y -- ยง 4. The generalized sieve problem -- ยง 5. The Viggo Brun method -- ยง 6. Selbergโs upper-bound method: informal discussion -- ยง7. Selbergโs upper-bound method -- ยง 8. Selbergโs lower-bound method -- ยง 9. Selbergโs lower-bound method: further discussion -- ยง 10. The โlargeโ sieves of Linnik and Rรฉnyi -- V. Primitive Sequences and Sets of Multiples -- ยง 1. Introduction -- ยง 2. Density -- ยง 3. An inequality concerning densities of unions of congruence classes -- ยง 4. Primitive sequences -- ยง 5. The set of multiples of a sequence: applications including the proofs of Theorems 4 and 5 -- ยง 6. A necessary and sufficient condition for the set of multiples of a given sequence to possess asymptotic density -- ยง 7. The set of multiples of a special sequence -- ยง 8. Proof of Theorem 15 -- ยง 2. The distribution of prime numbers -- ยง 3. Mean values of certain arithmetic functions -- ยง 4. Miscellanea from elementary number theory -- References -- Postscript -- Author Index