LECTURE NOTES OF WILLIAM CHEN
# HARDY-LITTLEWOOD METHOD

### Chapter 1 : WARING'S PROBLEM >>

### Chapter 2 : GOLDBACH'S PROBLEM >>

### Chapter 3 : DIOPHANTINE INEQUALITIES >>

### Chapter 4 : ROTH'S THEOREM ON ARITHMETIC PROGRESSIONS >>

This set of notes has been organized in such a way to create a single volume suitable for a brief introduction to one of the major developments of twentieth century mathematics, the famous Hardy-Littlewood method for studying additive problems in analytic number theory. We discuss sums of powers of integers, sums of primes, diophantine inequalities and conclude with a description of Roth's celebrated theorem on arithmetic progressions. The material represents a somewhat expanded version of selected chapters of the excellent monograph of RC Vaughan on the subject.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

- Introduction
- The Minor Arcs
- The Major Arcs
- The Singular Integral
- The Singular Series
- Weyl's Inequality and Hua's Lemma

- Introduction
- The Minor Arcs
- The Major Arcs
- Input from the Distribution of Primes
- A Fundamental Identity of Vaughan

- Introduction
- The Trivial Regions
- The Major Arc
- The Minor Arcs

- Introduction
- A Major Arc Type Argument
- Completion of the Proof