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.

To give you a brief idea of what you can expect to be able to do at the end of the course here are the Intended Learning Outcomes:

• prove elementary results on sums over primes and use these to calculate averages of
• utilise the correspondence between the product of Dirichlet series and convolution
of arithmetic functions to factor multiplicative functions and then calculate their
averages,
• prove some analytic properties of the Riemann zeta function, including an analytic
continuation, a zero-free region, and estimates on the growth of the zeta function,
• prove the Prime Number Theorem with an error term.

## UEQ: Feedback on feedback

I have made a few comments on the feedback you have given me and the course. As I have said in that, I am glad you have enjoyed the course, I enjoyed giving it. Feedback on UEQs

## The Timetable

• 3 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     MATH41022
Lecture
Alan Turing G.108
Except for week 8: G.207

Tuesday
Wednesday
Thursday
Friday         MATH41022
Lecture
Alan Turing G.209

## Lecture Notes

Notes Contents
Chapter 1
Appendix Chapter 1
Two proofs of the infinitude of primes. Definition of the Riemann zeta function, infinite products.
Chapter 2 part 1 Elementary Prime Number Theory. von Mangoldt's Function; Partial Summation; Replacing sums by integrals.
Chapter 2 part 2 Chebyshev's bounds for ψ (x), π (x) and θ (x), Asymptotic relations between ψ (x) & θ (x) and π (x) & θ (x).
Chapter 2 part 3 Merten's results on weighted sums over primes; extended with improved error terms; Merten's Theorem on Euler products.
Chapter 2 part 4 The Statement of the Prime Number Theorem, π(x)~ x/logx if, and only if, ψ(x)~ x, the logarithmic integral.
Chapter 2 part 5 Appendix
Chapter 2 Part 6 Summary
Graphs Graphs of π(x), x/logx and lix.
Chapter 3 Parts 1 Arithmetic functions. Cauchy Products; Convolutions; Dirichlet Series.
Chapter 3 Part 2 Multiplicative Functions. Möbius function; Möbius inversion.
Chapter 3 Part 3 Factorising arithmetic functions; Pertubations of known functions; Dirichlet Series as products and quotients of the Riemann zeta function; Euler’s phi function.
Summary Summary of Chapter 3.
Table Table of Arithmetic Functions, including their decompositions and associated Dirichlet Series.
Appendix 1 Appendix 1 Inverses of Arithmetic functions.
Appendix 2 Appendix 2 The Dirichlet Series for multiplicative functions has an Euler Product.
Chapter 4 part 1 Sums of Convolutions. Convolution method I.
Chapter 4 part 2 Sums of Convolutions. Convolution method II; average orders; the decomposition of d2.
Chapter 4 part 3 Sums of Convolutions. Various summations -- of Q2, 2ω, d(n2) and finally d2(n).
Chapter 4 Part 4 Sums of Convolutions. Dirichlet's hyperbolic method.
Table Table of Arithmetic Functions, their associated Dirichlet Series, and Summations; statements of possible improvements to Summation results proved in the course.
Appendix 1 Appendix 1 Convolution Method 3, the general case.
Appendix 2 Appendix 2 More on the Divisor Function.
Appendix 3 Appendix 3 Asymptotic Result on Summation of dk(n).
Appendix 4 Appendix 4 Extra terms in Summation results.
Chapter 5 Sums of Additive Functions. Turán's inequality; Hardy-Ramanujan Theorem.
Chapter 5 Appendix Appendix. Turán-Kubilius inequality; statement of the Erdos-Kac Theorem.
Chapter 6
Introduction
The Prime Number Theorem. Introduction
Step 1 Step 1, Appendix Step 1 Analytic Properties of the Riemann zeta function.
Step 2 Step 2, Appendix Step 2 Relating the prime counting function, ψ(x), to the Riemann zeta function.
Step 3 Step 3, Appendix Step 3 ζ(σ +it) ≠ 0 for σ ≥ 1.
Step 4 Step 4, Appendix

Step 4 Bounds on the Riemann zeta function.
i.   Upper bounds for σ ≥ 1 - η(t) for some function η(t) > 0,
ii.   Lower bounds for σ ≥ 1- φ(t) for some function φ(t) > 0.
Divergenceof ζ(1 +it).

Step 5 Step 5 Moving the line of integration.
Step 6 Step 6 Final deduction of the Prime Number Theorem.
Actual & Conjectured results on the zeros of ζ(s).

## Problem and Solution Sheets

Problem and Solution Sheets
Problem Sheet 1
1-15
Problem Sheet 2
1-18

Problem Sheet 2
19-28
Problem Sheet 3
1-21

Problem Sheet 3
Problem Sheet 4
1-8

Problem Sheet 4
Solution Sheet 1
1-15
Solution Sheet 2
1-18
Solution Sheet 2
19-28
Solution Sheet 3
1-6
Solution Sheet 3
7-14
Solution Sheet 3
15-18
Solution Sheet 3
19-21
Solution Sheet 3
Solution Sheet 4
1-8
Solution Sheet 4
9-12
Question Sheet 5
1-3
Question Sheet 6
1-12
Question Sheet 6
13-16

## Past & Present exam papers

Up to 2016 the level 4 and 6 version of the Analytic Number Theory course was an extension of the level 3 version. This was reflected in the exam where there was an extra question on the higher level material. From the 2016\17 academic year the higher level material is distributed throughout the course and it will be examined in all the questions of the exam.

Exam Papers Solutions and
Feedback
2014-15 Exam Solutions for 2014-15
2015-16 Exam Solutions for 2015-16
2016-17 Exam Solutions for 2016-17

## 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.

Background Notes Contents
Complex Analysis I

Complex Analysis I. What I expect you to know from years 1 and 2. There is no better place to look than in Charles Walkden's notes for MATH20101, Complex Analysis.

Complex Analysis II Complex Analysis II What I will take for granted but which you would not necessarily have seen in earlier years: Cauchy Sequences, uniform convergence, Weierstrass M-test, Weierstrass's Theorem for Sequences, Weierstrass's Theorem for Series, Weierstrass's Theorem for Infinite Integrals.
Analytic Continuation

Analytic Continuation.

Product of Series

Products of Series. Cauchy Product and Dirichlet Convolution of series.

Infinite Products Theory of Infinite Products. Conditions under which the infinite product ∏n=1(1+an) converges.
The big O notation The O and o notation. The O, o, ≪, ∼ and ≍ notations
Logarithmic Differentiation Logarithmic Differentiation

## 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 MATH41022. 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 MATH41022. 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 MATH41022. 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 MATH41022. 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.