**Colloquium Publications**

Volume: 57;
2010;
527 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-0-8218-4970-5

Product Code: COLL/57

List Price: $103.00

Individual Member Price: $82.40

**Electronic ISBN: 978-1-4704-1766-6
Product Code: COLL/57.E**

List Price: $103.00

Individual Member Price: $82.40

#### Supplemental Materials

# Opera de Cribro

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*John Friedlander; Henryk Iwaniec*

This monograph represents the state of the art
both in respect of coverage of the general methods and in respect of
the actual applications to interesting problems.

A unique feature of this monograph is how the authors take
great pains to explain the fundamental ideas behind the proofs and to
show how to approach a question in a correct fashion. So, this book
is not just another monograph useful for consultation; rather, it is a
teaching instrument of great value both for the specialist and the
beginner in the field.

The authors must be congratulated for this exceptional
monograph, the first of its kind for depth of content as well as for
the effort made to explain the ‘why’ and not limiting
themselves to the ‘how to’. This is a true masterpiece
that will prove to be indispensable to the serious researcher for
many years to come.

—Enrico Bombieri, Institute for Advanced Study

This is a truly comprehensive account of sieves and their applications, by two of the world's greatest authorities. Beginners will find a thorough introduction to the subject, with plenty of helpful motivation. The more practised reader will appreciate the authors' insights into some of the more mysterious parts of the theory, as well as the wealth of new examples. No analytic number theorist should be without this volume, but it will not have a place on my bookshelves—it will be permanently on my desk!

—Roger Heath-Brown, University of
Oxford,Fellow of Royal Society

This is a comprehensive and up-to-date treatment of sieve methods. The
theory of the sieve is developed thoroughly with complete and accessible
proofs of the basic theorems. Included is a wide range of applications,
both to traditional questions such as those concerning primes, and to
areas previously unexplored by sieve methods, such as elliptic curves,
points on cubic surfaces and quantum ergodicity. New proofs are given also
of some of the central theorems of analytic number theory; these proofs
emphasize and take advantage of the applicability of sieve ideas.

The book contains numerous comments which provide the reader with insight
into the workings of the subject, both as to what the sieve can do and
what it cannot do. The authors reveal recent developments by which the
parity barrier can be breached, exposing golden nuggets of the subject,
previously inaccessible. The variety in the topics covered and in the
levels of difficulty encountered makes this a work of value to novices and
experts alike, both as an educational tool and a basic reference.

#### Readership

Graduate students and research mathematicians interested in number theory.

#### Reviews & Endorsements

Our understanding of sieves has improved greatly over the past 20 or 30 years, in large part due to the efforts of this book's authors, and proofs that use to take many pages in an almost incomprehensible notation can now be done cleanly in one page. This book does a good job of keeping the notation under control.

-- MAA Reviews

Written by the two leading authorities on the subject, it contains a wealth of insights as well as a string of new results. ... The topics covered here are remarkably wide-ranging, as are the connections which the authors make. Indeed this book is not just a volume on sieves, but rather a treatise on analytic number theory more generally. As such it is recommended to everyone with an interest in the subject.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Opera de Cribro

- Cover Cover11
- Title page iii4
- Contents v6
- Preface xi12
- Sieve questions 122
- Elementary considerations on arithmetic functions 1334
- Bombieri’s sieve 2142
- Sieve of Eratosthenes-Legendre 3152
- Sieve principles and terminology 3556
- Brun’s sieve—The big bang 5576
- Selberg’s sieve—Kvadrater er positive 89110
- Sieving by many residue classes 139160
- The large sieve 151172
- Molecular structure of sieve weights 173194
- The beta-sieve 185206
- The linear sieve 235256
- Applications to linear sequences 259280
- The semi-linear sieve 275296
- Applications—Choice but not prime 305326
- Asymptotic sieve and the parity principle 331352
- Combinatorial identities 345366
- Asymptotic sieve for primes 355376
- Equidistribution of quadratic roots 373394
- Marching over Gaussian primes 383404
- Primes represented by polynomials 395416
- Level of distribution of arithmetic sequences 405426
- Primes in short intervals 441462
- The least prime in an arithmetic progression 453474
- Almost-prime sieve 475496
- Appendix A. Mean values of arithmetic functions 487508
- Appendix B. Differential-difference equations 507528
- Bibliography 519540
- Index 525546
- Back Cover Back Cover1554