Contents

Preface v

16 Limiting Distributions  553
16.1 Uniform distribution (mod 1)  553
16.2 Quantitative estimates  557
16.3 Kronecker’s theorem  565
16.4 Almost periodicity  571
16.5 The weak convergence of measures  573
17 Exponential sums — I. van der Corput’s method  581
17.1 Exponential integrals  581
17.2 Elementary estimates  584
17.3 van der Corput’s method  594
18 Estimates for sums over primes  617
18.1 Principles of the method  617
18.2 Applications
19 The large sieve
19.1 The analytic principle
19.2 Elementary formulations
19.3 Character sums
20 Primes in arithmetic progressions — III
20.1 Averages of |ψ(x, χ)|
20.2 The Bombieri–Vinogradov theorem
20.3 Mean square distribution
21 Additive Prime Number Theory
21.1 Sums of three primes
21.2 Sums of two primes on average
21.3 Small gaps between primes
21.4 Lower bounds for error terms
21.5 Conditional estimates
22 Mean and large values of Dirichlet polynomials
22.1 Mean value estimates
22.2 Approximate functional equations
22.3 Mean values of L-functions

23 Zero density theorems
24 Exponential sums — II. Vinogradov’s method
24.1 Vinogradov’s mean value theorem
24.2 Vinogradov’s method
25 Primes in short intervals
26 Linnik’s theorem
26.1 A sharper zero-density estimate
26.2 The Deuring–Heilbronn Phenomenon
26.3 The least prime in arithmetic progression
26.4 The irregular distribution of primes in short intervals
27 Probabilistic Number Theory
27.1 The Turán–Kubilius inequality
27.2 Mean values of multiplicative functions
27.3 The distribution of additive functions
27.4 Applications of probability theory
27.5 Multiplicative functions with vanishing mean value
28 Sieves II
28.1 The sieve of dimension 1
28.2 A weighted sieve estimate
28.3 Chen’s theorem

APPENDICES

E Further topics in harmonic analysis
E.1 Quantitative trigonometric approximation
F Bounds for Bilinear Forms
F.1 The operator norm of a matrix
F.2 Square matrices

Preface

This is a preliminary release to students in Math 775 of material intended for a much larger work (roughly 25 chapters in two volumes). Readers are urged to contact the authors concerning any misprints that are detected. Other comments or suggestions for improvement will also be very welcome.

The object of this book is to introduce the interested student to the techniques, results, and terminology of multiplicative number theory. It is not intended that our discussion will always reach the research frontier. Rather, it is hoped that the material here will prepare the student for intelligent reading of the more advanced research literature. vi