Analytic Number Theory 2020

The positive integers do not lie, like the logical foundations of mathematics, in the scarcely visible distance, nor in the uncomfortably tangled foreground, like the immediate data of the physical world, but a decent middle distance, where the outlines are clear and yet some element of mystery remains...There is no one so blind that he does not see them, and no one so sharp-sighted that his vision does not fail; they stand there a continual and inevitable challenge to the curiosity of every healthy mind. I have merely directed your attention for a moment to a few of the less immediately conspicuous features of the landscape, in the hope that I may sharpen your curiosity a little, and that some may feel tempted to walk a little nearer and take a closer view.

G. H. Hardy

This is the course page for the Part III course Analytic Number Theory, Lent Term 2020.

Examination and revision

This year's exam will be different in difficulty and length than usual, and will be Pass/Fail.
There will be three questions, and you are expected to attempt all three. You have three hours. For more details on the practical arrangements and the code of conduct, see the department Coronavirus page.
I have written two sample exam questions, to give an indication of the style of questions, which can be found here.
The exam is open book, so in particular you can have lectures notes with you in the exam (both your own and the PDF from this website). You are not allowed to access the internet during the exam, so you should download the PDF notes from this website in advance.
Since the syllabus changes yearly, not all exam questions from past years are suitable. Questions that I have identified as being covered by this year's syllabus are: 2019 (Q1 and Q3), 2014 (Q1a, Q2, Q3a, Q5), 2012 (Q3), and 2010 (Q1). Past exam papers are available from here. (The relevant course name changes slightly, but should be easily identifiable). Note that the length and difficulty of these past questions is not representative of what will be on this year's exam.
The revision class for this course on the 18th May will be recorded and available for download from Moodle.

Coronavirus-related update

As with all teaching, remaining teaching for this course in the Easter term will be held online.
There will be two more examples classes, covering Sheets 3 and 4. The first will be held on Monday 27th April between 2pm and 3:30pm. The second will be held on Monday 4th May between 2pm and 3:30pm.
There will also be a revision class, which will be held on Monday 18th May between 2pm and 3:30pm.
All online teaching will be carried out on Zoom. Shortly before each class a Zoom link and password will be emailed to everyone on the course mailing list. If you're not sure whether you're on this mailing list, please contact me at tb634@cam.ac.uk.

Lecture Notes: A PDF version of the lecture notes is available here. This is now fully updated, and should be a complete account of everything covered in lectures.

Problem sheets:

The first problem sheet is available here, now with solutions available here.
The second problem sheet is available here, now with solutions available here.
The third problem sheet is available here, now with solutions available here. A solution to the extra bonus question (that shows there are infinitely many zeros on the critical line) is available here.
The fourth problem sheet is available here, now with solutions available here

Analysis digest: I've written a short summary of the analysis, both real and complex, that will be used in this course. It attempts to give all the relevant definitions and theorems, and at least sketches of proofs, but is probably too brief to be any use in learning from. It should rather be viewed as a refresher, a reminder of things once known but now forgotten. It can be found here.

Books: There are many books on analytic number theory available. My favourite (which covers a lot of what is covered in this course, and great deal besides) is Multiplicative Number Theory, by Montgomery and Vaughan. The definitive reference for the classical theory of the Riemann zeta function is The Theory of the Riemann Zeta-Function by Titchmarsh. A shorter treatment can be found in Riemann's zeta function by Edwards. But what is better than the source: a translation of Riemann's original 1859 paper can be found here.