Students should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving results of suitable accessibility. See the minimal learning outcomes section of the Math 687R Wiki page for a clear indication of the level of difficulty appropriate to the course.

1. Riemann's memoir on the zeta function.
2. The functional equation of the L functions.
3. Properties of the gamma function.
4. Integral functions of order 1.
5. The infinite products for ξ(s) and ξ(s).
6. Zero free regions for ζ(s) and L(s, χ).
7. The counting functions N(T) and N(T, χ).
8. The explicit formula for ψ(x) and the prime number theorem.
9. The explicit formula for ψ(x, χ) and the prime number theorem for arithmetic progression.
10. Siegel's theorem and application to prime numbers in arithmetic progressions.
11. Vaughan's identity.
12. The large sieve.
13. The Bombieri-Vinogradov theorem.
14. The Barban-Davenport-Halberstram theorem.