# MA3001 Mastergradsseminar i matematikk høsten 2008

**Sessions** Monday 16:15-17:00 in Room 734 (SB II, Dept of Maths), Wednesdays 16:15-17:00 in Room 734

**Oral Examination on Wednesday the 10th of December, room 922 SB II, 13-17 o'clock**

**An essay to be delivered no later than 8.XII"**

First lecture on Monday the 8th of September at 16:15 in Room 734.

In the fall of 2008 the main topic is the Analytic Theory of Numbers. Thus the connection between the prime numbers and the celebrated zeta function of Riemann is a central theme. Prerequisities: some complex analysis. If interested, come to SB2, room 1152 on Thursday 4 September at 16:15.

Tentative book: *Tom Apostol* "Introduction to Analytic Number Theory

- Chapter 2 Arithmetical Functions and Dirichlet Multiplication

§ 2.1 - § 2.9

- Chapter 3 Averages of Arithmetical Functions

§ 3.1 -3.5, 3.7, 3.10, 3.11

- Chapter 4 Some Elementary Theorems on the Distribution on Prime Numbers

§ 4.1 -4.5, 4.8 Numerical example for the proof of Tschebyschef's Thm

- Chapter 6 Finite Abelian Groups and their Characters

§ At least 6.8, 6.9, and 6.10 Summation for B(x) and Interesting sums

- Chapter 7 Dirichlet's Theorem on Primes in Arithmetical Progressions

- Chapter 11 Dirichlet Series and Euler Products

Perhaps not § 11.9. In addition, Cahen's formulas for the abscissas.

- Chapter 12 The Functions ζ(s) and …

At least the Riemann zeta function.

- Chapter 13 Analytic Proof of the Prime Number Theorem

Also a proof without the Riemann-Lebesgue lemma (instead part of the contour is to the left of the abscissa 1; see Stein-Shakarchi: *Complex Analysis*)

- Zeros of the Riemann zeta functionZeros and Products