"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

**ATTENTION:** This is the course page for an old year. In particular, the course content has changed significantly, so the course notes and problem sheets should not be taken as a guide to the content of the course taught in 2020.

This is the course page for the Part III course Analytic Number Theory, Lent Term 2019.

- There will be 24 lectures, on Tuesdays, Thursdays, and Saturdays, at 11am in MR12.
- Example classes will be in Weeks 3, 5, and 7, Tuesdays from 3:30pm till 5pm, in MR14.
- Drop-in sessions will be held in Weeks 2, 4, 6, and 8, Tuesdays from 4pm till 5pm, in the central core.

**Lecture Notes:** A PDF version of the lecture notes can be found

__here__. These lecture notes should be a reasonably accurate representation of the entire course.

**Problem sheets:** Two questions from each sheet will be marked if handed in by 5pm on the Monday before an examples class. Work should be handed into the appropriate folder of the B pigeonhole in DPMMS by 5pm on Monday if you want it to be marked. Typed solutions for each sheet will be posted here a few days after the examples class.

**Books:** There are many books on analytic number theory available. My favourite (which includes everything covered in this course, and great deal besides) is

**Multiplicative Number Theory**, by Montgomery and Vaughan. A classic reference for the first part of the course, on elementary methods, is Hardy and Wright's

**An Introduction to the Theory of Numbers**.

**Complex analysis:** In the second half of the course basic knowledge of complex analysis will be assumed. A summary of all the facts that will be assumed prior knowledge is available

__here__. If something on that sheet seems unfamiliar, you should look it up in any first text on complex analysis (such as Priestly's Introduction to Complex Analysis, Lang's Complex Analysis, or Needham's Visual Complex Analysis).

**Exam and revision:** The exam for this course will be at 1:30pm on Thursday 6th June. There will be four questions on the paper, of which you are expected to answer three of your choice (the best three answers will count). There will be a revision class for this course on
Thursday 16th May, 3:30pm, in MR5.

**Past exam questions:** The content of Part III courses varies significantly from year to year, which means that there are not many previous exam papers suitable for practice. I have identified some questions from previous years that you should be able to tackle using what you have learnt in this course.
These come with the disclaimer that I had no involvement with these courses or exams, and they should not be taken as indicative of the style or content of questions in this year's exam, just as exam-style practice questions on the relevant material. Past year exam papers can be found

here.

- 2017, Modular Forms and L-Functions, Question 2.
- 2015, Elementary Methods in Analytic Number Theory, Questions 1 (parts a and b), 3, 5.
- 2012, Prime Numbers, Question 1.
- 2010, Analytic Number Theory, Questions 1, 3.
- 2006, Analytic Number Theory, Questions 1, 4, 5.