# MA3150 Analytic Number Theory, Spring 2021

### SCHEDULE FOR EXAM. (There can be some delay)

Tuesday 1 June
9.00 - 9.30 FREE SLOT
9.30 - 10.00 Lars Dalaker
…..
10.15 - 10.45 Ole Edstrøm
10.45 - 11.15 Gustav Bagger
…..
11.30 - 12.00 Oskar Goldhahn
12.00 - 12.30 Markus Hagen
…………………….
13.30 - 14.00 Tor Kringeland
14.00 - 14.30 Chileshe Mwamba.

Wednesday 2 June
9.00 - 9.30 Henrik Langesæter
9.30 - 10.00 Bendik Løvlie
…..
10.45 - 11.15 Marcel Sommerfelt ?
…..
11.30 - 12.00 Erling Svela
………………………
13.00 - 13.30 Espen Sørhaug
13.30 - 14.00 Thomas Thrane

### Lectures by Zoom, beginning 25 March

The lectures by zoom on April 7th and 8th are a repetition of the syllabus.

### Exercises below 2021

A letter from N. Abel Legendre's 1.08366

### x

Analytic number theory studies the distribution of the prime numbers, based on methods from mathematical analysis. Of central importance is the study of the Riemann zeta function, which embodies both the additive and the multiplicative structure of the integers. It turns out that the location of the zeros of this meromorphic function is intimately linked to the distribution of the primes. At the end of the nineteenth century, this insight led to the celebrated prime number theorem. The zeta function has been subject to intensive research ever since, but many fundamental questions remain open, of which the Riemann hypothesis undoubtedly is the most famous.

Key words for the course: Arithmetic and multiplicative functions, the Euler - Maclaurin summation formula, Abel summation and Möbius inversion, Dirichlet series and Euler products, the Riemann zeta function, the functional equation for the zeta function, the gamma function, (the Mellin transformation and Perron's formula), the prime number theorem, the Riemann hypothesis, Dirichlet characters, Dirichlet's theorem on primes in arithmetic progressions.

Prerequisites: Complex Analysis corresponding to what is covered by TMA4120 Matematikk 4K. In particular, contour integrals. Useful to know the Maximum Principle, the Principle of Argument Principle of Argument, the logarithm.

## Lecturer

Peter Lindqvist: Office 1152 in SB II, peter [dot] lindqvist [at] ntnu [dot] no

## Lectures

### From March 25th lectures by Zoom.

• Wednesday 10:15-12, F3
• Thursday 12:15-14, F3

The first lecture: January 14. The first lectures only on Zoom.

Meeting Id: 958 9577 5432 Passcode 161399

## Textbook

Harold Davenport, “Multiplicative Number Theory", Third Edition, Springer Graduate Texts in Mathematics. The lectures will to a be based on this book, but some supplementary material will also be considered. In particular, in the beginning of the course, we will pick some material from the first chapters of Tom M. Apostol, "Introduction to Analytic Number Theory", Springer Undergraduate Texts in Mathematics. Both books are available online from Universitetsbiblioteket.

## Syllabus and requirements for the examination

The syllabus for the course is as defined by the lectures described in detail below. I expect you to be able to present the basic concepts and ideas discussed during the lectures. The exercises should be viewed as an integral part of the syllabus.

* Plan of the exam.* Due to "force majeure" the original structure had to be modified.
The exam is oral and "physical".

1) First, you should present one of the topics below, using 15-20 minutes. IN NORWEGIAN OR ENGLISH. Which topic is decided during the exam, not in advance. (Thus it is wise to prepare them all.) Please, prepare for 15 minutes, in some of the topics only the main steps are possible within the available time, details can be left out.
* Tschebyshef's Theorem
* The functional equation for the zeta-function of Riemann
* The Hadamard Product for the xi-function.
* The line s = 1 + ti is zero-free.
* L(1,khi) is not zero for a complex character khi.
During the presentation you are allowed to use handwritten notes (one A4 sized sheet of paper, both sides available). Notice that the presentation (on the blackboard) does not have to contain more than the main points, but you may be asked about some detail.

2) Second, questions are asked and some basic example has to be solved.

Proper grades A, B, C, D, E, F are given, if possible.

## Reference Group

Gustav Bagger — gustavba@stud.ntnu.no
Markus Hagen — markus.v.hagen@ntnu.no

## Contents of the lectures (2021)

• Lecture 1, January 14: A short historical introduction with the Riemann zeta function $\zeta(s)$. Euler product for $\zeta(s)$ and Euler's proof of the divergence of the series of reciprocals of the primes (from Davenport Ch. 1). Then from Chapter 2 of Apostol's book. The Möbius function $\mu(n)$, Euler's totient function $\varphi(n)$; basic properties and the relation between these two functions, Dirichlet convolution, Möbius inversion, the von Mangoldt function $\Lambda(n)$, multiplicative functions.
• Lecture 2, January 20. T. Apostol: Sections 2.1-2.7, 2.8, 3.1 - 3.3. Alternative to 2.5 is 5.10, pp. 123-124.
• Lecture 3, January 21. T. Apostol Sections 3.1-3.5, 4.1-4.4.
• Lecture 4, January 27. T. Apostol Section 4.5 (Tchebyshef's estimates), Legendre's identity in 3.11 is needed, Section 4.8 (Mertens' theorem)
• Lecture 5, January 28. Poisson Summation (ref ?), The Gamma Function (p. 250 T. Apostol), Dirichlet Series (Apostol, 11.1-3
• Lecture 6, February 3. Dirichlet Series. 11.6, 11.7. Cahen's Formula for conv. abscissa.
• Lecture 7, February 4. Dirichlet series. 11.4, 11.8, 11.10, 11.11. zeta(1+it)
• Lecture 8, February 10. Estimates for zeta(s), zero free region. 13.4, 13.5
• Lecture 9, February 11. From Chapter 13. Notes. The contour Integral. A residue calculation
• Lecture 10, February 17. Proof of the Prime Number Theorem (crossing the line with abscissa 1 as in the books E. Stein, R. Shakarchi: Complex Analysis (Ch. 7) or A. Ingham: The Distribution of Prime Numbers. — Riemann's Functional Equation (Davenport, Chapter 8). Riemann's Functional Eqn.
• Lecture 11, February 18. Infinite Products, convergence. Some basic def.Weierstrass's product for entire functions.
• Lecture 12, February 24. Hadamard's thm for functions of order one, Jensen's thm. Hadamard's thm, order = 1. See also Davenport, Chapter 11.
• Lecture 13, February 25. The Weierstrass product for Xi(s) and Zeta(s). Growth order of Xi(s). Davenport, Chapter 12.
• Lecture 14, March 3. Zeros of Zeta(s)
• Lcture 15, March 4. More about the zeros. Riemann's Hypothesis ?
• Lecture 16, March 10. A zero free zone. Davenport Ch 13. Some exercises.
• Lecture 17, March 11. A zero free zone. Dirichlet's Thm.
• Lecture 18, March 17. Dirichlet characters, Apostol Ch. 6.8. Non-vanishing of L-functions. Apsostol Ch. 6.9.
• Lecture 19, March 18. Proof of Dirichlet's thm as in E. Stein - R. Sakarchi: 'Fourier Analysis', Chapter 8.—Exercises.
• Lecture 20, March 24. Some Geometry of Numbers. Minkowski's theorem. Some execises.
• Lecture 21, March 25. Quadratic Gaussian sums, an example. A proof of reciprocity
• Lecture 22, April 7. Repetition of Syllabus.
• Lecture 23, April 8. Repetition of Syllabus.
• Lecture 24, April 14. Repetition of Syllabus.
• Lecture 25, April 15, Another proof of Riemann's functional eqn, Bernoulli numbers. (Apostol, 12.8 but not via Hurwitz' zeta):Zeta
• Lecture 26, April 21. End of previous proof, Bernoulli numbers, zeta(2n) = …(Apostol 12.11, 12.12). Exercises.
• Lecture 27 , April 22. LAST LECTURE. Twilight zone: two strange theorems.—- If needed, comments on exercises.
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## Exercises

====2021==== Examples 1--5 .Some solutions
* EXERCISES Apostol: Chapter 1: 16, 17, 30. Chapter 3: 1, 2, 3. Chapter 4: 8, 18, 19 Chapter 11: 2, 15.
* EXERCISES Examples 6--11 Misprint in 10: read … -1| \leq 2|w|^2 (squared).
Exercises Examples 12--15
Some solutions Solutions. Solution of 11/3
* EXERCISES Text Misprint in 18: answer zeta(2)/zeta(3). The factor p+1 should be p-1 . Some solutions

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## Exam, dates and location

Oral examinations will take place June 1st and 2nd in Room KJL 22. See schedule on top of this page. Do not come late, but you may have to wait a little.

## Guidance and consultation before the exam

Send me an email first. If needed, Zoom can be used.