In the fall quarter (starting Sep 27) I will be teaching a graduate course on analytic prime number theory. This will be similar to a graduate course I taught in 2015, and in particular will reuse several of the lecture notes from that course, though it will also incorporate some new material (and omit some material covered in the previous course, to compensate). I anticipate covering the following topics:

- Elementary multiplicative number theory
- Complex-analytic multiplicative number theory
- The entropy decrement argument
- Bounds for exponential sums
- Zero density theorems
- Halasz’s theorem and the Matomaki-Radziwill theorem
- The circle method
- (If time permits) Chowla’s conjecture and the Erdos discrepancy problem [Update: I did not end up writing notes on this topic.]

Lecture notes for topics 3, 6, and 8 will be forthcoming.

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4 September, 2019 at 2:40 pm

AnonymousThe new material (topics 3, 6, 8) seems to be really new (recent research material !)

4 September, 2019 at 3:14 pm

rafik zeraouliaIs there any link to follow your courses ?

4 September, 2019 at 4:56 pm

WesleyHow can I enroll??

5 September, 2019 at 3:59 am

John MangualIs this separate from your higher-order Fourier analysis course?

11 September, 2019 at 7:09 am

Steffen YountThis material is almost all stuff I’m really interested in!

I wish I could audit this course!

Is there any chance you could record the lectures and make the course publicly available online?

12 September, 2019 at 1:27 pm

U1. Why is the entropy decrement argument a topic to study?

2. Would Diophantine objects and discrepancy be covered?

13 September, 2019 at 7:04 am

Terence TaoThe entropy decrement argument can be used to establish a number of cases of (the logarithmically averaged version of) Chowla’s conjecture, particularly when combined with the Matomaki-Radziwill theorem and the circle method; but there are some more recent applications of the method that do not require these additional tools (for instance, my student has recently used this method to obtain a new proof of the prime number theorem), so I have decided to give it a separate set of notes (particularly since the use of that method requires a review of the Shannon entropy inequalities, which are rather separate from the other techniques in this set of notes.

I will not cover Diophantine approximation or discrepancy in these notes (and indeed will be omitting several other core topics in analytic number theory, notably sieve theory). Montgomery’s “Ten lectures in harmonic analysis” is still quite a good general reference for this subject.

24 September, 2019 at 8:25 pm

GrendHow does the entropy decrement argument end up leading to a proof of the prime number theorem, roughly? Does it allow one to get around dealing with zero-free regions?

21 September, 2019 at 12:07 pm

plmDear Terence Tao, I hope you will finally decide to publish your analytic number theory notes -in the GSM series. I always thought they were some of your most valuable expository writings, especially because of the growing gap between the research front and textbook material on number theory -too often very introductory, elementary, and not presenting modern philosophies. Don’t hesitate to add even the most fleeting intuition, all the mental simplifications that guide experts but are scattered throughout litterature, if expressed at all. Thank you very much.

12 October, 2020 at 10:51 am

RyanDear Terry,

were the notes for 3, 6, and 8 ever completed?

Warmly,

Ryan

[Links updated -T.]