## Textbook

There is no required textbook for the course. Instead I'll place the following books on reserve at the library:

1. G.J.O. Jameson, The prime number theorem. Cambridge 2003 LMS Student texts vol 53.

2. T. Apostol, Introduction to analytic number theory, Undergraduate texts in Mathematics, Springer Verlag, 1976.

3. H.L. Montgomery and R.C. Vaughan, Multiplicative number theory, I. Classical theory. Cambridge university press, Cambridge studies in advanced math vol 97, 2007.

4. Mendes-France and Tenenbaum, The prime numbers and their distribution, American Math. Soc. Student Math. Library 6, 2000.

5. J. Stopple, A primer of analytic number theory, Cambridge 2003.

6. R. Ayoub, An introduction to the Analytic theory of numbers. AMS 1963.

7. H. Davenport, Multiplicative number theory. Springer GTM.

Also I'll put up notes on this website. My aim in this course will be to discuss several problems related to the distribution of prime numbers. One high point for the course will be the proof of the prime number theorem which gives an asymptotic for the number of primes up to x. I will also discuss Riemann's seminal paper introducing the zeta function as a tool in prime number theory, explain some of the properties of zeta, and the connection between primes and the Riemann hypothesis. Other topics may include sieves (e.g. showing upper bounds for the number of twin primes), the primality test, gaps between primes etc. I will assume that you have some knowledge of number theory already, at the level of 152, and that you're also comfortable with analysis and thinking about the size of things, and have some familiarity with complex analysis (say, up to Cauchy's theorem). If you're concerned with the background, please feel free to talk to me. To brush up on complex analysis you could look at the book by Green and Krantz (Function theory of one complex variable, first four chapters), or Ahlfors (Complex Analysis, first four chapters), or Copson (An introduction to the theory of functions of a complex variable, first six chapters).

Here are some notes based on lectures from Math 152 in previous years: Bertrand's postulate, Dirichlet's Theorem I, Dirichlet's Theorem II, Dirichlet's Theorem III, Dirichlet's Theorem IV. I won't be assuming that you know everything here, but familiarity (or comfort) with some of the techniques mentioned here will be useful, and I'll review some of this in the first couple of lectures.

Your grade will be based on several homework assignments (30%), one Midterm (30%), and a Final exam (40%). .

## Office Hours

I'll be available from 2:00-4:00 on Tuesday afternoons; my office is in 383W. You can also email me and schedule an appointment.

## Midterm

We will have an in class midterm exam on Thursday, May 3.

## Final Exam

The final exam will be in Room 381T of the Math building, from 3:30 to 6:30 PM on Friday, June 8.

## Homework Assignments

Homework will be assigned every Thursday, and will be due the following Thursday in class. No late submissions will be accepted; instead I will drop your lowest score.

Homework 1 due April 12.

Homework 2 due April 19.

Homework 3 due April 26.

Homework 4 due May 3.

Homework 5 due May 10.

Homework 6 due May 17.

Homework 7 due May 24.

Homework 8 due May 31.

## Notes

Moor Xu has been live-TeX-ing the course lectures, and has kindly made them available for all. Thanks, Moor!