Distribution of Prime Numbers

Distribution of Prime Numbers

by WWL Chen

This set of notes, previously known as Elementary and Analytic Number Theory, has been used between 1981 and 1990 by the author at Imperial College, University of London.

The material has been organized in such a way to create a single volume suitable for an introduction to the distribution of prime numbers.

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.

SECTION A --- ELEMENTARY ASPECTS

Chapter 1: ARITHMETIC FUNCTIONS (last uploaded on 2 December 2002)

Introduction
The Divisor Function
The Moebius Function
The Euler Function
Dirichlet Convolution

Chapter 2: ELEMENTARY PRIME NUMBER THEORY (last uploaded on 2 December 2002)

Euclid's Theorem Revisited
The Von Mangoldt Function
Tchebycheff's Theorem
Some Results of Mertens

SECTION B --- ANALYTIC ASPECTS

Chapter 3: DIRICHLET SERIES (last uploaded on 2 December 2002)

Convergence Properties
Uniqueness Properties
Multiplicative Properties

Chapter 4: PRIMES IN ARITHMETIC PROGRESSIONS (last uploaded on 2 December 2002)

Dirichlet's Theorem
A Special Case
Dirichlet Characters
Some Dirichlet Series
Analytic Continuation
Proof of Dirichlet's Theorem

Chapter 5: THE PRIME NUMBER THEOREM (last uploaded on 2 December 2002)

Some Preliminary Remarks
A Smoothing Argument
A Contour Integral
The Riemann Zeta Function
Completion of the Proof

Chapter 6: THE RIEMANN ZETA FUNCTION (last uploaded on 2 December 2002)

Riemann's Memoir
Riemann's Proof of the Functional Equation
Entire Functions
Zeros of the Zeta Function
An Important Formula
A Zero-Free Region
Counting Zeros in the Critical Strip
An Asymptotic Formula
The Prime Number Theorem