# What should be learned in an introductory analytic number theory course?

Hello all --

I have the privilege of teaching an introductory graduate course in analytic number theory at the University of South Carolina this fall. What topics should I definitely cover?

I'm not lacking for good material of course. I intend to cover much of Davenport; there is also Cojocaru and Murty's introduction to sieve methods; there is interesting elementary work by Chebyshev et al. on counting primes; there is also Apostol's excellent book; I could dip into Pollack's new book; and there are many other excellent sources as well. I should also make sure the students master partial summation, big-O, and the kinds of contour integration that come up in typical problems.

I feel prepared to do a good job, and I will also have good people to ask for advice in my new department, but I would cheerfully welcome further advice, opinions, etc. from anyone who would like to offer them. Any thoughts?

Thanks to all. --Frank

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Perhaps another complete syllabus might be valuable. Here is the one for MIT's 18.785, Spring 2007: ocw.mit.edu/courses/mathematics/… . – Qiaochu Yuan Jul 28 '11 at 2:49
@quid: typically for a grad course, 3 hours a week x 13 weeks. – Igor Rivin Jul 28 '11 at 2:54
@Frank: There are a lot of peoples syllabi/notes on the Number Theory Web: numbertheory.org/ntw/lecture_notes.html – Micah Milinovich Jul 28 '11 at 3:05
Igor, thank you; this is exacty in the middle of my two guesses. Alon, I assume 'Not always buried deep' by Paul Pollack is meant. – user9072 Jul 28 '11 at 6:15
Considering the title of the question, I suspect "Apostol's excellent book" in your question is his intro to analytic number theory (a UTM) rather than his book on modular functions and Dirichlet series (a GTM). If that is right, your graduate students might think it's kind of funny (not in a ha-ha way) if you base a graduate course on a book pitched at undergraduates. – KConrad Jul 28 '11 at 15:47

There are so many possible "first graduate courses in analytic number theory" that their mutual intersection must be very small.

After reflecting a bit, the two things which seem indispensable to me are some treatments of the Prime Number Theorem and Dirichlet's Theorem on Primes in Arithmetic Progressions. Combining these two things into one thing, I strongly recommend that you cover The Prime Number Theorem for Arithmetic Progressions. Of course Davenport spends more time on this than any other single topic in his book, so I'm sure you were not dreaming of skipping this. But that means it's the right answer, no?

I think the next "don't even think of skipping this" result is Dirichlet's Analytic Class Number Formula. (Let me say that this was not covered in any course I took as an undergraduate or graduate student nor any course that -- to the best of my knowledge -- was even offered.)

After these big theorems, I would make sure to spend some time developing the skills of analytic number theory, especially estimating various things in various ways. I have also always felt cheated not to have been taught Euler-Maclaurin summation (or even been made aware of its existence!), but I'm pretty sure that's not required. Summation by parts is a must, of course.

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If you do the class number formula, you've gotta do Goldfeld's proof of Siegel's bound: it's in Iwaniec & Kowalski, and is rather shorter than the proof in Davenport. – David Hansen Jul 28 '11 at 4:57
In the spirit of Pete's comment on the class number formula, prove some stuff that needs algebraic number theory, not just things that need only the language of Q and Dirichlet characters. – KConrad Jul 28 '11 at 15:44
One might relate the class number to the sum of three squares, mathoverflow.net/questions/3596/… – Will Jagy Jul 28 '11 at 18:48
But do the students have a little background in alg. num. thry? If they have, you can prove the Hecke theta formula (assuming Poisson summation) and use it to prove (1) the Dirichlet Unit Theorem, (2) finiteness of the class number, (3) the functional equation of the Dedekind zeta function, (4) the analytic class number formula, (5) the Dirichlet density of primes that split completely in a normal extension K of $\mathbb{Q}$ is $1/n_K$, and (6) the rational primes that split completely in a normal extension of $\mathbb{Q}$ determine it, in 14 pages with full proofs. – Marius Overholt Jul 28 '11 at 19:50
Another worthy goal is the Prime Ideal Theorem of Landau. Specializing this to $\mathbb{Q}$ yields the Prime Number Theorem in the form $\pi(x) \sim x/\log(x)$. A complete proof of the Prime Ideal Theorem by means of Ikehara's Tauberian theorem takes only a couple of pages. And the road to Ikehara's theorem is easier than it used to be, after Korevaar found a proof by means of Newman's contour integration method. With full details, this proof takes six pages, and does not require any Fourier analysis (but if you want the strongest form, you need the Riemann-Lebesgue Lemma.) – Marius Overholt Jul 28 '11 at 19:51

Discuss some applications of the generalized Riemann hypothesis to problems that are not at first directly about zeta or L-functions. For example, the Solovay-Strassen test leads to a polynomial-time primality test if GRH is true for all Dirichlet L-functions (well, you "just" need GRH for even characters). Of course Agrawal-Kayal-Saxena later gave an unconditional proof of that polynomial-time result, but I think the technique by which Solovay-Strassen would create a polynomial-time primality test is a nice illustration of analytic methods.

Moreover, when you do give applications of GRH, you ought to indicate what would happen in the theorem if one knew a uniform zero-free region of the form Re(s) > 1/2 + epsilon for some epsilon in (0,1/2). Many applications of GRH "only" need a common zero-free region in the critical strip, not the optimal common zero-free region Re(s) > 1/2 (at the expense of worse constants when epsilon > 0 instead of epsilon = 0).

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I just taught last semester a course on analytic number theory for 4th year undergraduate students. That was 2 hours a week during 14 weeks; the students had complex analysis before. I had a chapter on Dirichlet's theorem on primes in arithmetic progressions, a chapter on the prime number theorem (proof with a tauberian theorem), and a chapter on some aspects of Riemann's paper (two proofs of the analytic continuation of $\zeta$, the functional equation, the trivial zeroes, the special values).

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I have no idea how enlightened your students are, but you cannot go wrong with Vinogradov's little book "Elements of number theory". If your students are very strong, you will be done with it in a couple of weeks, and can go on to Davenport. If not, you can spend the whole term on it, and they will be wiser.

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Someone downvoted this (not me). I am not familiar with the book. Explanations for the downvotes? – Frank Thorne Jul 28 '11 at 21:16
I didn't downvote it, but I am familiar with the book. It is one of the nicer of the traditional introductory treatments of elementary number theory, up to quadratic reciprocity and primitive roots. The strong point of the book is a large number of both theoretical problems and computational exercises, with about 90 pages of solutions and answers at the back. This book may be the finest resource available for someone who wants to learn elementary number theory on their own. A weak point is that there is no index. I imagine the answer was downvoted because there is no analysis there. – Marius Overholt Jul 29 '11 at 8:42
@Marius: it is false that there is no analysis there. A big chunk of the book is devoted to estimating sums of arithmetically relevant functions (sometimes via integrals, Euler-Maclaurin-style, though those names are not mentioned), which was what Vinogradov did for a living (exponential sums). Since analytic number theory has been at times described as the art of summing, this is extremely relevant to anyone who wants to study analytic number theory. – Igor Rivin Jul 29 '11 at 14:24
@Igor: Well, there is a brief chapter on arithmetic functions, without any estimates. But in the section of problems following this chapter, there is a selection of estimates to prove. It certainly is not a big chunk of the book. Maybe we are not speaking of the same book? I am only familiar with the translated edition, and perhaps there was more material in the Russian edition. – Marius Overholt Jul 30 '11 at 10:14

Having recently finished my math undergrad I audited a course based on Davenport and had a reading course using Apostol, which to use depends on what skills and smaller results you want them to come away with. The things you mentioned like big-O and summation are given a pretty thorough treatment in Apostol. I certainly wasn't cheated and really appreciated having some practice with the skills. It also had enough material for you to have some flexibility.

But, it might seem like too much of an undergraduate text (it introduces the definition of a group before it talks about characters). It also gives a pretty elementary proof for Dirichlet's theorem. Which you may not want.

A book not mentioned that also has a lot of topics and is nice to learn from is Additive Number Theory by Melvyn Nathanson. The material here is very different from that of the other two, but still worthwhile and accessible.

If I had to pick one, I'd go with Apostol. It was so readable and I felt like I got a great foundation in the ideas and skills of number theory.

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This may be a stretch, but what about the Burgess bound for character sums? The Riemann hypothesis over finite fields is a ubiquitous tool in number theory now, and this seems as a good an introduction to it as any. Besides, if you're willing (as I would be!) to simply quote the key estimate for complete character sums, the deduction of Burgess from that is basically a clever application of Hölder's inequality and rearrangements, which of course are equally ubiquitous techniques.

My personal one-sentence summary of (most of) analytic number theory is roughly "Poisson summation, Holder's inequality, combinatorial rearrangement, and the Riemann hypothesis over finite fields form a noncommutative monoid", and I feel it would be ideal if students got a glimpse of all four of these techniques in an introductory course.

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Cool stuff. I think it will be overly ambitious for the main body of the course, but I have in mind a few seminar-style lectures at the end where I go off the deep end a little bit. This might be good for that! – Frank Thorne Jul 30 '11 at 2:48

Besides Davenport's book, which basically everyone has already recommended, why not talk a little more about the circle method? If I recall correctly, the only application Davenport gives is for counting the prime number solutions to an equation, but the circle method is probably simpler to understand when you're just looking for integer solutions. One can use Davenport's other book "Analytic Methods for Diophantine Equations and Inequalities". At the end, he also does the Oppenheim conjecture for 5 variables, which is somewhat simpler than the rest of the book beyond Waring's problem.

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I think Iwaniec and Kowalski is probably a cleaner reference for an introduction to the modern point of view on the circle method. (That really is one of the most amazing books.) – David Hansen Jul 29 '11 at 4:24
I agree that IK is fabulous, and on the circle method in particular. Nevertheless I hated it as a beginner. If I embolden anyone to crack its covers after my course is finished, I shall have succeeded spectacularly. – Frank Thorne Jul 29 '11 at 7:33
IK is not an introduction, more of a compendium. It does have everything there, but good luck finding it.. – Igor Rivin Jul 29 '11 at 14:24