Sieve Methods

Sieve Methods

HEINI HALBERSTAM

and

HANS-EGON RICHERT

DOVER PUBLICATIONS, INC.

Mineola, New York

Copyright

Copyright © 1974, 2011 by Heini Halberstam and the Estate of Hans-Egon Richert.

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2011, is an unabridged republication of the work originally published in 1974 by Academic Press, Inc., London. Heini Halberstam has provided a new Preface and a new Errata List for this edition.

Library of Congress Cataloging-in-Publication Data

Halberstam, H. (Heini)

Sieve methods / Heini Halberstam and Hans-Egon Richert.

p. cm.

Originally published: London; New York: Academic Press, 1974.

Includes bibliographical references and index.

eISBN-13: 978-0-486-32080-9

1. Sieves (Mathematics) I. Richert, H.-E. (Hans-Egon), 1924- II. Title.

QA246.H35 2011

512.7’.3—dc22

2010039623

Manufactured in the United States by Courier Corporation

47939001

www.doverpublications.com

Preface to the Dover Edition

There is never a right time to publish a systematic account of a subject that is in a state of healthy growth; even as the late Ted Richert and I were embarking on the writing of Sieve Methods forty years ago, inspired by the 1965 seminal of Jurkat and Richert, the young Henryk Iwaniec was discovering what has become known as the Rosser-Iwaniec method, which has since lead to many advances and applications, by himself and others. There is now in the literature an enveloping sieve (Hooley), a vector sieve (Brudern and Fourvy), a prime detecting sieve (Harman); and probably there will be others.

There are now also other books about sieves that the interested reader might wish to consult: there is the influential discussion of sieves by one of the founders of the subject, Atle Selberg, in Volume II of his Collected Papers (1991), there is Sieves in Number Theory (2001) by the late George Greaves, there is Glyn Harman’s monograph on Prime Detecting Sieves (2007), and recently Harold Diamond and I have published A Higher Dimensional Sieve Method (2009), building on Iwaniec’s approach. Several current books on Number Theory also devote sections or chapters to sieves.

Our book has by now been long out of print, but demand for it continues and second-hand copies, when they can be found, command exorbitant prices. I am very grateful therefore to Dover Publications for reprinting it. I apologize for the long list of Errata; that they were necessary in the original was entirely my fault.

Heini Halberstam

March, 2010

Preface

We conceived the idea of writing an introduction to applicable sieve theory during a brief encounter at Syracuse airport in the spring of 1966; we had both been lecturing on sieves, and it seemed to us that the time for a systematic exposition was long overdue. Our original plan was modest: to to supply, in lecture notes form, the theoretical background for the method of Jurkat-Richert [1] and to illustrate it by means of significant applications; also, to give a first connected account of the large sieve. As we burrowed in the literature and came to realize its extent and wealth—the notes we prepared on some two hundred papers in the summer of 1968 alone would have filled a monograph—the inadequacy of our original conception became increasingly clear. Not only did we find that we should have to assimilate much classical material into a revised scheme but, as we progressed, new results began to appear which affected it profoundly; so much so that, viewing the matter in retrospect, if we had called a halt at any earlier stage and published then, the book would have been much the poorer. In the end we abandoned the chapters on the large sieve altogether—happily the monographs of Montgomery [2] and Huxley [2] have appeared in the meantime and provide admirable introductions to that subject and much else beside—and concentrated on the small sieves of Brun and Selberg; even so, critics should have no trouble in pointing to omissions, and we accept any such criticisms in advance. Indeed, we have been at pains to point out important extensions that time and space forced us to exclude, and we hope only that what we have done here will make it easier for others to cover these too in due course.

The intermittent opportunities we have had of working together on the book have often shared the nature of that first encounter; even so, they would have been much rarer still but for the generous support of the National Science Foundation and of the U.S. Army Research Office, and for the equally generous hospitality of Syracuse University during several happy summers of uninterrupted work. We express our due gratitude to these institutions for all their help. We owe a special word of thanks also to Dr. R. C. Vaughan of Imperial College, London, for having read with great care several chapters of the manuscript and for having removed some blemishes that had escaped our notice; also for having pointed out to us a notable simplification in Chapter 11. The final version of the manuscript was typed by Mrs. Shoshana Mandel of Tel Aviv University and Mrs. Angela Fullerton of Nottingham, and we are indebted to them both for undertaking a task that must often have been extremely trying. We thank the London Mathematical Society for accepting the book into their series of research monographs. Finally, we are grateful to Academic Press for all their help and patience during the course of publication; Mr. Hitchings, of Roystan Printers Ltd., told us that our manuscript had been typographically the most difficult he had met in twenty years of printing mathematics.

The companionship of a shared endeavour has been a memorable experience for us both. Now that we have completed, with a mixture of sadness and relief, the task we set ourselves eight years ago, we hope that there will be some readers for whom our book will prove to be a helpful introduction to a beautiful topic that has been shrouded for too long in unnecessary mystery.

July, 1974

H. Halberstam

H. -E. Richert

Contents

Preface to the Dover Edition

Preface

Notation

Errata

Introduction

1.Hypotheses H and HN

2.Sieve methods

3.Scope and presentation

1: The Sieve of Eratosthenes: Formulation of the C problem

1.Introductory remarks

3.Basic examples

and the sifting function S

5.The sieve of Eratosthenes–Legendre

2: The Combinatorial Sieve

1.The general method

2.Brun’s pure sieve

3.Technical preparation

4.Brun’s sieve

5.A general upper bound O-result

6.Sifting by a thin set of primes

7.Further applications

8.Fundamental Lemma

9.Rosser’s sieve

3: The Simplest Selberg Upper Bound Method

1.The method

2.The case ω(d) = 1, |Rd| ≤ 1

4.The Brun–Titchmarsh inequality

5.The Titchmarsh divisor problem

7.The prime twins and Goldbach problems: explicit upper bounds

8.The problem ap + b = p′: an explicit upper bound

4: The Selberg Upper Bound Method (continued): O-results

1.A lower bound for G(x, z)

2.Applications

5: The Selberg Upper Bound Method: Explicit Estimates

1.A two-sided Ω2-condition

2.Technical preparation

3.Asymptotic formula for G(z)

4.The main theorems

5.Two ways of dealing with polynomial sequences {F(p)}: discussion

6.Primes representable by polynomials

7.Primes representable by polynomials F(p): the non-linearized approach

8.Prime k-tuplets

9.Primes representable by polynomials F(p): the linearized approach

6: An Extension of Selberg’s Upper Bound Method

1.The method

2.An upper estimate

3.The function σK

4.Asymptotic formula for G(ξ, z)

5.The main result

7: Selberg’s Sieve Method (continued): A First Lower Bound

1.Combinatorial identities

2.An asymptotic formula for S

3.Fundamental Lemma

4.The function ηK

5.A lower bound

6.The main result

8: TheLinear Sieve

1.The method

2.The functions F, f

3.An approximate identity for the leading terms

4.Upper and lower bounds for S

5.The main result

9: A Weighted Sieve: TheLinear Case

1.The method

2.Application to the prime twins and Goldbach problems

3.The weighted sieve in applicable form

4.Almost-primes in intervals and arithmetic progressions

5.Almost-primes representable by irreducible polynomials F(n)

6.Almost-primes representable by irreducible polynomials F(p)

10: Weighted Sieves: The General Case

1.The first method

2.The first method in applicable form

3.Almost-primes representable by polynomials

4.The second method

5.Almost-primes representable by polynomials

6.Another method

11: Chen’s Theorem

1.Introduction

2.The weighted sieve

3.Application of Selberg’s upper sieve

4.Transition to primitive characters

5.Application of contour integration

6.Application of the large sieve

Bibliography

References

Notation

STANDARD NOMENCLATURE

[x] always denotes the largest integer not exceeding x.

Where no confusion is possible with the notation for intervals, (a, b) denotes the highest common factor of integers a, b and [a, b] their lowest common multiple.

We denote the Möbius function by μ(n) (for a definition see p. 34). We write v(n) for the number of distinct prime divisors of n, Ω(n) for the number of prime divisors of n counted according to multiplicity, τ(n) for the number of positive integer divisors of n, and φ(n) for Euler’s function (the order of the multiplicative group of reduced residue classes modulo n).

We denote the number of primes not exceeding x by π(x) and the number of primes not exceeding x that are congruent to l modulo k by π(x; k, l).

SIEVE NOTATION

are defined on pages 24, 73 respectively.

The following table indicates where the various principal functions featuring in our account of sieves are first introduced:

The unique real root of the equation ηκ(u) = 1, written vκ, is introduced on page 212.

All our general theorems have been formulated subject to certain basic conditions. The symbolic representations of these conditions, and the pages on which they are first described, are as follows:

CONSTANTS

The symbols π and e have their usual meaning throughout, and γ always denotes Euler’s constant.

The letters A, B, C with or without suffices or superscripts denote constants ≥ 1 throughout. The A-constants, together with the constants κ and α, feature in the basic conditions and we therefore refer to them sometimes as structure constants. The dependence of the B- and C-constants is first described at the end of Chapter 1, Section 4 (pp. 29, 30), and there are also reminders at appropriate stages throughout the text.

The O- and o-notation of I. M. Vinogradov, are now so well established as not to require definition; for the dependence of the constants implied by the use of these notations see top of p. 30.

CONVENTIONS

We often use: = as the symbol for equality by definition; see e.g. p. 15.

Theorem 5.4 is the fourth theorem in Chapter 5; Corollary 5.4.2 is the second corollary of Theorem 5.4. Lemmas are numbered in the same way.

In any one chapter we attach to a formula the notation (3.6) if it is the sixth numbered formula in Section 3 of that chapter. If we wish in Chapter 7 to refer to formula (3.6) of Chapter 5 we do so by referring to it as (5.3.6).

To avoid excessive length in the statement of a general theorem (some statements are very long as it is!) we use a short-hand way of indicating which of the basic conditions the theorem rests on; for a description see footnotes on pp. 31, 57.

ABBREVIATIONS

RH denotes Riemann’s conjecture concerning the location of zeros of the Riemann ζ-function; GRH denotes the generalized Riemann conjecture, concerning the location of zeros of L-functions.

H-W stands for Hardy and Wright, An Introduction to the theory of numbers, 4th edition, Oxford 1960. D stands for Davenport’s Multiplicaative Number Theory, Markham, Chicago 1967. We refer to these texts for various classical results.

ERRATA

The following corrections should be made in the text:

p.1, l.6: for from some point onward read greater than 2

p.9, NOTES l.1: derives from Schinzel