An Introduction to Sieve Methods and Their Applications ( London Mathematical Society Student Texts )

Publication series ：London Mathematical Society Student Texts

Author： Alina Carmen Cojocaru; M. Ram Murty  

Publisher： Cambridge University Press‎

Publication year： 2005

E-ISBN: 9780511131493

P-ISBN(Paperback): 9780521848169

Subject： O156 Number Theory

Keyword： 数论

Language： ENG

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An Introduction to Sieve Methods and Their Applications

Description

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 an introductory graduate course in analytic number theory, and non-experts can gain a quick introduction to the techniques of the subject.

Chapter

Cover

Half-title

Series-title

Title

Copyright

Dedication

Contents

Preface

1 Some basic notions

1.1 The big ‘O’ and little ‘o’ notation

1.2 The Möbius function

1.3 The technique of partial summation

1.4 Chebycheff’s theorem

1.5 Exercises

2 Some elementary sieves

2.1 Generalities

2.2 The larger sieve

2.3 The square sieve

2.4 Sieving using Dirichlet series

2.5 Exercises

3 The normal order method

3.1 A theorem of Hardy and Ramanujan

3.2 The normal number of prime divisors of a polynomial

3.3 Prime estimates

3.4 Application of the method to other sequences

3.5 Exercises

4 The Turán sieve

4.1 The basic inequality

4.2 Counting irreducible polynomials in Fp[x]

4.3 Counting irreducible polynomials in Z[x]

4.4 Square values of polynomials

4.5 An application with Hilbert symbols

4.6 Exercises

5 The sieve of Eratosthenes

5.1 The sieve of Eratosthenes

5.2 Mertens’ theorem

5.3 Rankin’s trick and the function Psi(x, z)

5.4 The general sieve of Eratosthenes and applications

5.5 Exercises

6 Brun’s sieve

6.1 Brun’s pure sieve

6.2 Brun’s main theorem

6.3 Schnirelman’s theorem

6.4 A theorem of Romanoff

6.5 Exercises

7 Selberg’s sieve

7.1 Chebycheff’s theorem revisited

7.2 Selberg’s sieve

7.3 The Brun–Titchmarsh theorem and applications

7.4 Exercises

8 The large sieve

8.1 The large sieve inequality

8.2 The large sieve

8.3 Weighted sums of Dirichlet characters

8.4 An average result

8.5 Exercises

9 The Bombieri–Vinogradov theorem

9.1 A general theorem

9.2 The Bombieri–Vinogradov theorem

9.3 The Titchmarsh divisor problem

9.4 Exercises

10 The lower bound sieve

10.1 The lower bound sieve

10.2 Twin primes

10.3 Quantitative results and variations

10.4 Application to primitive roots

10.5 Exercises

11 New directions in sieve theory

11.1 A duality principle

11.2 A general formalism

11.3 Linnik’s problem for elliptic curves

11.4 Linnik’s problem for cusp forms

11.5 The large sieve inequality on GL(n)

11.6 Exercises

References

Index

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