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
Best Answer
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.