# Analytic Number Theory, fall 2008

NEW: Change in schedule! Lecture moved from 5/12 to Thursday 4/12, 10-12, P1145!

www-page: http://www.math.uu.se/~astrombe/analtalt08/analtalt08.html

Teacher: Andreas Strömbergsson email: firstname.lastname at math dot uu dot se (with "o" in place of "ö").

Literature:

Davenport: "Multiplicative Number Theory", Springer Verlag.
Lecture notes. (I may post lectures here before they are actually delivered, but please note that I may make changes, corrections and additions to those sections without any notification. However, if I make a non-trivial correction in the notes for a lecture which I have already given, I will post it HERE.)

Problems: See the end of each section in the lecture notes.
Assignment 1: Problems 2.3, 2.5, 3.2, 3.5, 4.6, 4.8, 6.5. (5p each; total 35p.) Due 3/10.
Assignment 2: Problems 7.4 (10p), 8.8 (15p), 10.1 (10p). (Total 35p.) Due 5/11.
Assignment 3: Problems 15.2 (5p), 16.2 (5p), 18.5 (15p), 19.2 (5p), 20.2 (5p). (Total 35p.) Due 5/12.
For the oral exam: Things to learn.
Please email me individually to decide a time for the oral exam! I prefer the dates 15 or 17 December, or later.

Total grade: This will be a combination of your grade on the home assignments and your "grade" on the final exam, roughly with the weights 75% (assignments), 25% (final exam). Note that the final exam will be an oral exam. (There was originally a written exam scheduled for 12 December, 9.00-14.00, but this is canceled.)

For inspiration: poster (in Swedish)

Syllabus (=kursplan) Schedule.

Preliminary plan of lectures:

Date Time Place Section Topic
1. Wed, 3/9 10-12 11167 Ch. 1,4 Primes in arithmetic progressions
2. Fri, 5/9 13-15 11167 handout Infinite products
3. Fri, 12/9 13-15 11167 handout Dirichlet series and partial summation
4. Wed, 17/9 10-12 11167 Ch 2-5 More on Dirichlet characters and L(1,chi)
5. Fri, 19/9 13-15 P2214 Ch 6 Dirichlet's class number formula
6. Wed, 24/9 13-15 P2145 Ch. 7 + handout The distribution of the primes
7. Wed, 1/10 10-12 P2215 handout The Prime Number Theorem
8. Fri, 3/10 13-15 11167 Ch. 10,11 The Gamma function; Integral functions of order 1
9. Wed, 8/10 13-15 11167 Ch 8-9 The functional equation for zeta(s) and L(s,chi)
10. Fri, 10/10 13-15 11167 Ch 12 The infinite products for xi(s) and xi(s,chi)
11. Wed, 15/10 10-12 11167 Ch. 13 (and 14) A zero-free region for zeta(s) (and L(s,chi))
12. Wed, 22/10 10-12 P2215 Ch. 14 Zero-free regions for L(s,chi)
13. Wed, 29/10 10-12 11167 Ch. 15-16 The numbers N(T) and N(T,chi)
14. Wed, 5/11 10-12 11167 Ch. 17-18 The explicit formula for psi(x)
15. Fri, 7/11 13-15 P2215 Ch. 19 The explicit formula for psi(x,chi)
16. Wed, 12/11 10-12 P2215 Ch. 20 The Prime Number Theorem for Arithmetic Progressions (I)
17. Fri, 14/11 13-15 11167 Ch. 21-22 Siegel's Theorem
18. Wed, 19/11 10-12 11167 Ch. 23 The Polya-Vinogradov Inequality
19. Fri, 21/11 8-10 P2215 Ch. 24-25 Further Prime Number Sums
20. Wed, 26/11 10-12 11167 Ch. 26 Sums of Three Primes PRESENTATION
21. Wed, 3/12 10-12 11167 Ch. 27 The Large Sieve
22. Thu, 4/12 10-12 P1145 Ch. 28 Bombieri's Theorem
Oral exam: Date and time is decided individually.