MA3001-TMA4310 Advanced topics in the study of the Riemann zeta function

“One of the millennium problems in Mathematics is the famous Riemann hypothesis. This hypothesis asserts that the non-trivial zeros of the Riemann zeta function have real parts equal to ½. A correct solution to this problem results in a 1 million dollars prize being awarded by the CLAY institute.”

In this course, we will discuss many things related to the Riemann zeta-function, the Riemann hypothesis and its consequences, and the reason that we cannot win, until now, these 1 million dollars.

This course is a natural continuation of Analytic Number Theory (MA3150). The course of Analytic number theory is recommended, but not strictly necessary. Instead, it will be strongly recommended to read Chapter 3 of the book of Tom M. Apostol (Introduction to Analytic number theory). Minimum background in analysis is necessary, as residues theorem in Complex Analysis, the Fourier transform, Lebesgue measure, Dominated convergence theorem, Fubini’s theorem, and Lp spaces. The form of teaching will be lectures, as well as problem sessions.

Lecturer

Messages

  • 23/08 First meeting to decide the schedule: Monday 23/08, 11am.
  • 23/08 The course will be presencial.
  • 23/08 Schedule of the course:
    Mondays 15:15-17:00,
    Thursdays, 14:15-16:00,
    Fridays, 10:15-11:00 (or 12:00).
  • 23/08 The room will be announced soon.
  • 23/08 This first week we will have class only on Thursday.
  • 26/08 The room for the first lecture will be the Seminar room 734: Gløshaugen Sentralbygg 2 7. etasje Seminarrom 734 (322_734).
  • 30/08 Today the class is canceled.
  • 02/09 This week we don't have classes.
  • 03/09 News about the 3rd week:
    -The class on Monday 06 of September will be online. I will lecture this class, using zoom. Here is the link:
    https://NTNU.zoom.us/j/92139246758?pwd=YnQxcDMxQVJCRHNYVjZwUG1tM0d3dz09
    ID de reunión: 921 3924 6758
    Código de Acceso: 392329
    -The class on Thursday 09 of September will be presencial. The lecture will be given by Kristian Seip in room 734 (seminaroom).
    -The exercises time on Friday 10 of September will be online. I will lecture this class, using zoom and using the same link of Monday.
  • 13/09 The room for the lecture will be Seminar Room 734 at 15:15 pm.
  • 17/09 Exercise time day by zoom, 16:00pm-18:00pm (send me an email before).
  • 20/09 The reference group is:
    - Sarah May Instanes: sarahmin@stud.ntnu.no;
    - Markus Valås Hagen: markus.v.hagen@ntnu.no.
  • 08/10 Exercise time day by zoom, 10:15am-11:00am.
  • 27/10 Tomorrow the lecture will be by zoom:
    https://NTNU.zoom.us/j/92139246758?pwd=YnQxcDMxQVJCRHNYVjZwUG1tM0d3dz09
    ID de reunión: 921 3924 6758
    Código de acceso: 392329
  • Office hours: Tuesday 07/12, 14:00pm-16:00pm.
  • Final exam (oral): It will be on Monday 13/12 in the Seminar room 734. The schedule of the exam is:
    - William Tell: 10:00am - 10:40am;
    - Tomas Agung: 10:45am - 11:25am;
    - Sarah May: 11:30am -12:10pm;
    - Markus Valas: 1:20pm - 2:00pm;
    - Gustav Bagger: 2:05pm - 2:45pm;
    - Lars Dalaker: 2:50pm - 3:30pm.

Lecture notes and problem lists

1. Problem set 1
2. Problem set 2
3. Problem set 3
4. Problem set 4
5. Paper: Bounding zeta on the Riemann hypothesis
6. Paper: Some extremal functions in Fourier analysis II

Extra solutions of the exercises

* Solution_19_Sarah and Markus
* Solution_29e
* Solution_31
* Solution_33

Textbook

  • The Theory of the Riemann zeta-function, E. C. Titchmarsh, revised by D. R. Heath-Brown, Second edition, Clarendon Press Oxford.
  • Multiplicative number theory, H. Davenport, Second edition, Graduate Texts in Mathematics 74, Springer-Verlag, New York (1980).
  • Lectures on the Riemann zeta function, H. Iwaniec, University Lecture Series Volume 62, American Mathematical Society.

Syllabus

We will try to cover the following topics:
1. The Riemann zeta-function: History, definition, free-region zeros in Re(s)>1.
2. Analytic extension of the Riemann zeta-function at Re(s)>0; Integration by parts.
3. Free zeros in Re(s)=1.
4. Analytic continuation of zeta.
5. Hadamard product and especial constants.
6. Free-region zeros in the critical strip.
7. The Riemann hypothesis.
8. The number of zeros N(T).
9. The argument function S(T) and S1(T): bounds.
10. Gaps between zeros.
11. Montgomery-Vaughan inequality.
12. Mean values theorems of zeta.
13. Bounds for zeta (unconditional).
14. Percentage of zeros in the critical line: Hardy & Littlewood, Selberg-Levinson.
15. Guinand-Weil explicit formula.
16. Bounds for zeta conditional.
17. Extreme values conditional.

Classroom

First week: The Riemann zeta function: Definition

  • 26/08 Class1: The Riemann zeta function in Re(s)>1.

Second week: Canceled

Third week: The Riemann zeta function: analytic extension

Fourth week: Zeros-free region of zeta

Fifth week: Zeros of the Riemann zeta function

Sixth week: The approximation formula

Seventh week: Guinand-Weyl explicit formula

  • 04/10 Canceled by vaccination
  • 07/10 Class10: Guinand-Weyl explicit formula (Prof. Kristian Seip)
    Video: Video_zeta_class7

Eighth week: Montgomery Vaughan inequality

Ninth week: Behavior of the Riemann zeta-function I

Tenth week: Behavior of the Riemann zeta-function II

Eleventh week: Behavior of the Riemann zeta-function II

Twelfth week: Zeros of zeta

Thirteenth week: Zeros on the critical line

Fourteenth week: Zeros on the critical line and more conjectures

2022-02-03, carlosch