The official source of material for this course is the syllabus page, but here I will give more details of the course and its examination procedures.

For MMath and MSc material please press this button
level 4 and 6 material

## The Timetable

• 2 Lectures per week
• 1 example classes per week.
9.00 10.00 11.00 12.00 1.00 2.00 3.00 4.00 5.00
Monday MATH3\4\61022
Lecture
G.209
MATH4\61022
level 4 Lecture
G.110

Tuesday
Wednesday
Thursday
Friday         MATH3\4\61022
Lecture & tutorial
Alan Turing G.209

## Course Information.

This course can be seen as the use of Real and Complex Analysis in the study of numbers - particularly prime numbers. To do this course you must be comfortable with Mathematical Analysis and thus you should have done well in courses such as MATH10242 Sequences and Series, MATH20111 Real Analysis, MATH20101 Real and Complex Analysis and MATH20142 Complex Analysis.

The material covered by level 4 students will be the same as for the level 3 students, but to a greater depth. So the higher level students will see and be expected to understand a number of proofs not seen by the level 3 students. This is especially true when we prove the Prime Number Theorem.
Throughout the course the level 4 students will get more and harder questions on the problem sheets. This fits in with the School philosophy that level 4 students should spend at least 7 hours, outside lectures, on study for a course unit while for level 3 students it is a minimum of 4 hours.
You need to realise that taking the level 4 version of this course is not an easy way in which to get an extra 5 credits.

## Past & Present exam papers

There have been two previous papers for this course.

 2010-11 Exam Solutions for 2010-11 2011-12 Exam Solutions for 2011-12

## Lecture Notes

The lecture notes will be made available shortly after the material has been presented in lectures. The number of lectures are in brackets [...].

Notes Contents
Chapter 1

Two proofs of the infinitude of primes. [2.5]
Definition of the Riemann zeta function, infinite products.

Chapter 2

Elementary Prime Number Theory. [5]
von Mangoldt's Function. Chebyshev's bounds on the prime counting function, Partial Summation, Replacing sums by integrals, Merten's results on weighted sums over primes.

π(x)~ x/logx if, and only if, ψ(x)~ x.

Appendix: Graphs

Graphs of π(x), x/logx and lix.

Chapter 3 Intro The Prime Number Theorem. [6]
Chapter 3

Analytic Properties of the Riemann zeta function.

Relating the prime counting function to the zeta function.

ζ(σ +it) ≠ 0 for σ ≥ 1.

Bounds on the zeta function.

Moving the line of integration.

The Riemann-Lebesgue Lemma.

Final deduction of the Prime Number Theorem

Appendix
Appendix

The Riemann - Lebesgue Lemma

Chapter 4

Arithmetic functions and Dirichlet Series. [5]
Cauchy Products, Convolutions, Multiplicative Functions, Möbius function, Möbius inversion, Writing Dirichlet Series as products and quotients of the Riemann zeta function. Factorising arithmetic functions.

Table

Table Table of Arithmetic Functions, including their decompositions and associated Dirichlet Series.

Appendices

Appendix Dirichlet Series equals a Euler product

Chapter 5 Sums of arithmetic functions. [2]
Convolution method, and applications to the sums of the arithmetic functions Q2 the characteristic function of square-free integers, d the divisor function, d3=1*d, 2ω, d(n2), d2(n) and finally Euler's phi function φ(n).

## Problem and Solution Sheets

Problem and Solution Sheets
Problem Sheet 1

Problem Sheet 1
Level 4

Problem Sheet 1
Problem
Sheet 2 v2
Problem
Sheet 3
Problem
Sheet 4
Problem
Sheet 5
Soln Sheet 1

Soln Sheet 1
level 4

Soln Sheet 1
Solns 1-11

Solns 12-15

Solns 16-20

Solns 21-31
Solns 1-11

Solns 12-15

Solns 16 & 17
Solns 1-6

Solns 7-14

Solns 15-21

Solns 22-34
All Solns

## The Analysis in Analytic Number Theory

Here I have extracted the analysis that will be seen in Analytic Number Theory. Thus you will see little mention of Number Theory. Before you register for this course make sure that you are happy with, or can reasonably imagine that you become happy with, the material here.

You can download all the files in one LARGE FILE, or in parts below. There are problems for you to try in these files and the solutions will be given in the feedback classes for the course.

Background Notes Contents
Part 1

Replacing Sums by Integrals I. Integral Test for Series.

n=2 1n(log n)σ    converges if, and only if, σ > 1.

1N   ≤   ∑Nn=1 1n   −   logN   ≤   1

Part 2 Products of Series. Cauchy Product and Dirichlet Convolution of series.
Part 3

Summing Series. For example, how to sum 1+4y+9y2+16y3+25y4+... or sum 1+2y+4y2+6y3+8y4+10y5+...

Part 4 Theory of Infinite Products. Conditions under which the infinite product ∏n=1(1+an) converges.
Part 5 Replacing Sums by Integrals II. Abel or Partial Summation, Euler's Summation Formula, Stirling's formula.
Part 6 The O and o notation.
Part 7 Analysis. Cauchy Sequence, holomorphic and analytic functions, uniform convergence, Weierstrass M-test, Weierstrass's Theorem for Sequences, Weierstrass's Theorem for Series, Weierstrass's Theorem for Infinite Integrals.
Part 8 Poles, Zeros and Logarithmic Differentiation.
Part 9 Fourier Series and the function B1({x}) = x-[x]-1/2
Part 10 Fourier Transforms

## Recommended Texts

I have looked at a number of books in designing this course. These are listed below with a few sentences on each.

I would hope that my notes are self-contained, but if can not follow my approach to a subject you might look in the books below to find an alternative approach that might appeal to you more.

[A] T. Apostol, Introduction to Analytic Number Theory, 1st edition. 1976, Corrected 5th edition 2010, Springer, 1441928057
This is probably the best reference for the material on Arithmetic functions, sums of such functions and elementary prime number theory.

[D]   H. Davenport, revised by H.L. Mongomery, Multiplicative Number Theory, 2nd edition, Springer, 1980, 0-387-90533-2.
This is another classic Analytic Number Theory text, though at too high a level for MATH31022. Read it to see what the follow on course would have been.

[EWG. Everest, T. Ward, An Introduction to Number Theory, Graduate Texts in Mathematics 232, Springer, 2005, 1-85233-917-9.
Chapter 8 has a useful discussion on the Riemann Zeta function; with careful attention paid to questions of the where the function is holomorphic.

[HWG.H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, Oxford Science Publications, 1983, 0-19-853171-0.
This is a classic reference for Number Theory. Chapters XVI - XVIII are an excellent source for material on Arithmetic Functions, while Chapter XXII has a lot of material on the elementary Prime Number Theory. The book also contains an elementary proof of the Prime Number Theorem which is beyond the scope of this course.

[IJH. Iwaniec, E. Kowalski, Analytic Number Theory, AMS Colloquium Publications, Vol. 53, AMS 2004, 0-8218-3633-1.
This is a huge book of 610 pages where the first 42 cover more than is in MATH31022. You should read this to get a feel of where the subject has gone in the years after the proof of the Prime Number Theorem.

[J]  G.J.O. Jameson, The Prime Number Theorem, LMS Student Texts 53, CUP 2003, 0-521-89110-8.
This is a major reference source for the final chapters of MATH31022. The book contains two approaches to the Prime Number Theorem, of which we only study one. And in fact, just at the end of the proof of the proof of the PNT we switch to the approach in [T] below.

[NW. Narkiewicz, The Development of Prime Number Theory, Springer Monographs in Mathematics, Springer, 2000, 3-540-66289-8.
This gives an excellent historical perspective on the development of Prime Number Theory, but should be read more in the way of background reading.

[NZMI. Niven, H.S. Zuckerman, H.I. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley, 1991, 9-971-51301-3.
This book is a very well judged book for undergraduate Number Theory. For us, Chapter 4.3 contains Mobius Inversion while Chapter 8 discusses Elementary Prime Number estimates. Be careful, the book discusses Dirichlet Series but only for real s.

[SSW. Schwarz, J. Spilker, Arithmetic Functions, LMS Lecture Note Series 184, CUP, 1994, 0-521-42725-8.
As the title suggests, this book will tell you more about arithmetic functions than you may ever want to know. For us, only sections 1.1 - 1.4 are of interest.

[SGG. Sansone, J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable I. Holomorphic Functions, P. Noorhoff, Ltd Groningen, 1960.
This is simply a reference for results from Complex Analysis, there should be plenty of alternatives on the library shelves.

[TG. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, 46, CUP, 1995, 0-521-41261-7.
An excellent source for all things Analytic Number Theory, and thus the book goes far further than we can in MATH31022. The only reservation is that Dirichlet Series are done as Laplace-Stieltjies transforms, which is too advanced an approach for us. I finish the proof of the Prime Number Theorem by following p.169 of this book.

[THBE.C. Titchmarsh, revised by D.R. Heath-Brown, The Theory of the Riemann Zeta-function, 2nd edition, Oxford Science Publications, 1986, 0-19-853369-1.
This is a classic reference for results on the Riemann Zeta function, but apart from the first few pages it has little for us. It should be read for background, and though it was written in 1951, Heath-Brown has written new appendices to each Chapter describing what has been proved in the 35 years since first publication.