LECTURE NOTES OF WILLIAM CHEN

 

DISTRIBUTION OF PRIME NUMBERS

The first six chapters of this set of notes, previously known as Elementary and Analytic Number Theory, have been used between 1981 and 1990 by the author at Imperial College, University of London. Chapter 7 has been added in 2013.

The material has been organized in such a way to create a single volume suitable for an introduction to the distribution of prime numbers. Chapters 1 and 2 cover basic elementary techniques but do not reach the Prime number theorem. We then introduce analytic techniques in Chapters 3 - 6, and establish some of the classical results in the subject. We finally return to elementary technqiues to discuss Selberg's elementary proof of the Prime number theorem.

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.

SECTION A --- ELEMENTARY ASPECTS

Chapter 1 : ARITHMETIC FUNCTIONS >>

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

Chapter 2 : ELEMENTARY PRIME NUMBER THEORY >>

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

SECTION B --- ANALYTIC ASPECTS

Chapter 3 : DIRICHLET SERIES >>

• Convergence Properties
• Uniqueness Properties
• Multiplicative Properties

Chapter 4 : PRIMES IN ARITHMETIC PROGRESSIONS >>

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

Chapter 5 : THE PRIME NUMBER THEOREM >>

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

Chapter 6 : THE RIEMANN ZETA FUNCTION >>

• 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

SECTION C --- ELEMENTARY ASPECTS AGAIN

Chapter 7 : ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM >>

• Selberg Inequalities
• A Smoothing Argument
• A Logarithmic Viewpoint
• Deduction of the Prime Number Theorem