# MA3001 Mastergradsseminar i matematikk høsten 2011

**First lecture on Thursday the 8th of September in room 734, 14–16 o'clock.**

**Lectures** Monday 14–15 room 922 SB II; Thursday 14–16 room 734 SB II.

**Oral Examination Thursday 24 and Friday 25 of November.**

**An essay to be delivered no later than 23 of November**

In the fall of 2011 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. There will be an information meeting in room 1152 SBII at 15.15 o'clock on Friday 2. september.

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*)

- Functions of Growth Order One Products

- Zeros of the Riemann zeta functionZeros and Products