You are here

The Large Sieve and Its Applications: Arithmetic Geometry, Random Walks and Discrete Groups

Emmanuel Kowalski
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Tracts in Mathematics 175
[Reviewed by
Allen Stenger
, on

This book might be subtitled, "101 Uses for the Large Sieve". That is its strength and its weakness: it shows the broad gamut of applications for the large sieve (going far beyond it origins in number theory), but any particular reader may well be interested in only a few of these.

The Brun sieve and the Selberg sieve are considered to be small sieves, in the sense that they can only sieve out a bounded number of residue classes even when the modulus grows. The large sieve can increase the number of excluded residue classes as the modulus grows, and is very useful for developing results on averages. The best-known application of the large sieve is the Bombieri-Vinogradov theorem, that states that on average the error term in the prime number theorem for arithmetic progressions is small. Under the Generalized Riemann Hypothesis, the error term is always small, and the Bombieri-Vinogradov theorem can often be used as a substitute for the (still unproved) Generalized Riemann Hypothesis.

I found the exposition in this book hard to follow, mostly because of its generality. The large sieve has a number of formulations, even within number theory, and most of them do not look like sieves at all. In order to cover the wide variety of applications that the author wants to showcase, it's necessary to derive an extremely general formulation of the large sieve.

The present book has a strong algebraic flavor, and most of the applications are for counting various types of objects related to groups, especially several types of special linear and general linear groups. There's also a fair amount on Galois groups and some coverage of zeta functions of curves over finite fields. One of the more concrete applications is an asymptotic result for the number of primes dividing the denominators of rational points on an elliptic curve.

If your interest is primarily analytic number theory, this is not the book for you. Davenport's Multiplicative Number Theory has an extremely clear development of the large sieve and uses it to prove the Bombieri-Vinogradov theorem. Cojocaru & Murty's An Introduction to Sieve Methods and Their Applications gives numerous number-theoretic applications of the large sieve.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.



Prerequisites and notation

1 Introduction
1.1 Presentation
1.2 Some new applications of the large sieve

2 The principle of the large sieve
2.1 Notation and terminology
2.2 The large sieve inequality
2.3 Duality and 'exponential sums'
2.4 The dual sieve
2.5 General comments on the large sieve inequality

3 Group and conjugacy sieves
3.1 Conjugacy sieves
3.2 Group sieves
3.3 Coset sieves
3.4 Exponential sums and equidistribution for group sieves
3.5 Self-contained statements

4 Elementary and classical examples
4.1 The inclusion-exclusion principle
4.2 The classical large sieve
4.3 The multiplicative large sieve inequality
4.4 The elliptic sieve
4.5 Other examples

5 Degrees of representations of finite groups
5.l Introduction
5.2 Groups of Lie type with connected centres
5.3 Examples
5.4 Some groups with disconnected centres

6 Probabilistic sieves
6.1 Probabilistic sieves with integers
6.2 Some properties of random finitely presented groups

7 Sieving in discrete groups
7.1 Introduction
7.2 Random walks in discrete groups with Property (τ)
7.3 Applications to arithmetic groups
7.4 The cases of SL(2) and Sp(4)
7.5 Arithmetic applications
7.6 Geometric applications
7.7 Explicit bounds and arithmetic transitions
7.8 Other groups

8 Sieving for Frobenius over finite fields
8.1 A problem about zeta functions of curves over finite fields
8.2 The formal setting of the sieve for Frobenius
8.3 Bounds for sieve exponential sums
8.4 Estimates for sums or Betti numbers
8.5 Bounds for the large sieve constants
8.6 Application to Chavdarov's problem
8.7 Remarks on monodromy groups
8.8 A last application

Appendix A Small sieves
A.I General results
A.2 An application

Appendix B Local density computations over finite fields
B.I Density of cycle types for polynomials over finite fields
B.2 Some matrix densities over finite fields
B.3 Other techniques

Appendix C Representation theory
C.1 Definitions
C.2 Harmonic analysis
C.3 One-dimensional representations
C.4 The character tables of GL(2, Fq) and SL(2, Fq)

Appendix D Property (T) and Property (τ)
D.1 Property (T)
D.2 Properties and examples
D.3 Property (τ)
D.4 Shalom's theorem

Appendix E Linear algebraic groups
E.1 Basic terminology
E.2 Galois groups of characteristic polynomials

Appendix F Probability theory and random walks
F.1 Terminology
F.2 The Central Limi! Theorem
F.3 The Borel-Cantelli lemmas
F.4 Random walks

Appendix G Sums of multiplicative functions
G.1 Some basic theorems
G.2 An example

Appendix H Topology 
H.1 The fundamental group
H.2 Homology
H.3 The mapping class group of surfaces