London Mathematical Society Monographs

Glyn Harman

• Publisher: Princeton University Press
• Year: 2012
• ISBN:
  9781400845934
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

This book seeks to describe the rapid development in recent decades of sieve methods able to detect prime numbers. The subject began with Eratosthenes in antiquity, took on new shape with Legendre's form of the sieve, was substantially reworked by Ivan M. Vinogradov and Yuri V. Linnik, but came into its own with Robert C. Vaughan and important contributions from others, notably Roger Heath-Brown and Henryk Iwaniec. Prime-Detecting Sieves breaks new ground by bringing together several different types of problems that have been tackled with modern sieve methods and by discussing the ideas common to each, in particular the use of Type I and Type II information.
No other book has undertaken such a systematic treatment of prime-detecting sieves. Among the many topics Glyn Harman covers are primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes, and the recent work of John Friedlander and Iwaniec on primes that are a sum of a square and a fourth power, and Heath-Brown's work on primes represented as a cube plus twice a cube. This book contains much that is accessible to beginning graduate students, yet also provides insights that will benefit established researchers.

Table of contents

• Cover
• Contents
• Preface
• Notation
• Chapter 1. Introduction
• Chapter 2. The Vaughan Identity
• Chapter 3. The Alternative Sieve
• Chapter 4. The Rosser-Iwaniec Sieve
• Chapter 5. Developing the Alternative Sieve
• Chapter 6. An Upper-Bound Sieve
• Chapter 7. Primes in Short Intervals
• Chapter 8. The Brun-Titchmarsh Theorem on Average
• Chapter 9. Primes in Almost All Intervals
• Chapter 10. Combination with the Vector Sieve
• Chapter 11. Generalizing to Algebraic Number Fields
• Chapter 12. Variations on Gaussian Primes
• Chapter 13. Primes of the Form x[sup(3)] + 2y[sup(3)]
• Chapter 14. Epilogue
• Appendix
• Bibliography
• Index