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