TITLE: What should be learned in an introductory analytic number theory course?
QUESTION [21 upvotes]: 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

REPLY [2 votes]: 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.