(Ebook) An Introduction to Sieve Methods and Their Applications 1st Edition by Alina Carmen Cojocaru, M Ram Murty ISBN 9780511131493 0511131496

(Ebook) An Introduction to Sieve Methods and Their Applications 1st Edition by Alina Carmen Cojocaru, M Ram Murty ISBN 9780511131493 0511131496

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ISBN 10: 0511131496
ISBN 13: 9780511131493
Author: Alina Carmen Cojocaru, M Ram Murty

(Ebook) An Introduction to Sieve Methods and Their Applications 1st Edition Table of contents:

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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Tags: Alina Carmen Cojocaru, M Ram Murty, Sieve Methods, Applications