An introduction to sieve methods and their applications / Alina Carmen Cojocaru, M. Ram Murty.

Format:

Book

Author/Creator:

Cojocaru, Alina.

Contributor:

Murty, Maruti Ram.

London Mathematical Society.

Series:

London Mathematical Society student texts ; 66.

London Mathematical Society student texts ; 66

Language:

English

Subjects (All):

Sieves (Mathematics).

Physical Description:

xii, 224 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge, UK ; New York : Cambridge University Press, 2005.

Summary:

Sieve theory has a rich and romantic history. The ancient question of whether there exist infinitely many twin primes (primes p such that p+2 is also prime), and Goldbach's conjecture that every even number can be written as the sum of two prime numbers, have been two of the problems that have inspired the development of the theory.

This book provides a motivated introduction to sieve theory. Rather than focus on technical details, which can obscure the beauty of the theory, the authors focus on examples and applications, developing the theory in parallel. The text can be used for a senior level undergraduate course or for an introductory graduate course in analytic number theory, and non-experts can gain a quick introduction to the techniques of the subject.

Contents:

1 Some basic notions 1

1.1 The big 'O' and little 'o' notation 1

1.2 The Mobius function 2

1.3 The technique of partial summation 4

1.4 Chebycheff's theorem 5

2 Some elementary sieves 15

2.2 The larger sieve 17

2.3 The square sieve 21

2.4 Sieving using Dirichlet series 25

3 The normal order method 32

3.1 A theorem of Hardy and Ramanujan 32

3.2 The normal number of prime divisors of a polynomial 35

3.3 Prime estimates 38

3.4 Application of the method to other sequences 40

4 The Turan sieve 47

4.1 The basic inequality 47

4.2 Counting irreducible polynomials in F[subscript p] [x] 49

4.3 Counting irreducible polynomials in Z[x] 51

4.4 Square values of polynomials 53

4.5 An application with Hilbert symbols 55

5 The sieve of Eratosthenes 63

5.1 The sieve of Eratosthenes 63

5.2 Mertens' theorem 65

5.3 Rankin's trick and the function [Psi] (x, z) 68

5.4 The general sieve of Eratosthenes and applications 70

6 Brun's sieve 80

6.1 Brun's pure sieve 81

6.2 Brun's main theorem 87

6.3 Schnirelman's theorem 100

6.4 A theorem of Romanoff 106

7 Selberg's sieve 113

7.1 Chebycheff's theorem revisited 113

7.2 Selberg's sieve 118

7.3 The Brun-Titchmarsh theorem and applications 124

8 The large sieve 135

8.1 The large sieve inequality 136

8.2 The large sieve 139

8.3 Weighted sums of Dirichlet characters 142

8.4 An average result 147

9 The Bombieri-Vinogradov theorem 156

9.1 A general theorem 157

9.2 The Bombieri-Vinogradov theorem 167

9.3 The Titchmarsh divisor problem 172

10 The lower bound sieve 177

10.1 The lower bound sieve 177

10.2 Twin primes 185

10.3 Quantitative results and variations 193

10.4 Application to primitive roots 195

11 New directions in sieve theory 201

11.1 A duality principle 201

11.2 A general formalism 205

11.3 Linnik's problem for elliptic curves 207

11.4 Linnik's problem for cusp forms 209

11.5 The large sieve inequality on GL(n) 213.

Notes:

Includes bibliographical references (pages 218-221) and index.

ISBN:

0521848164

0521612756

OCLC:

60794238

Publisher Number:

9780521848169 (hbk.)

9780521612753 (pbk.)