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

Coronavirus-related update

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:

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.