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
 A: 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). 
A: 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.
A: 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.
A: 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.
A: 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.  
A: 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).
A: 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.
