Analytisk tallteori våren 2014

(MA3001 Mastergradsseminar i matematikk)

In the spring semester of 2014, a main topic of MA3001 will be Analytic Number Theory. Our main goal will be to understand the connection between the prime numbers and the celebrated Riemann zeta function. Prerequisites: Complex Analysis corresponding to what is covered by TMA4120 Matematikk 4K. There will be a first meeting in room 734 in SB2 on January 7 at 10:15 a.m. A main purpose of this meeting will be to fix 4 weekly hours for lectures and exercises.

Book: Tom Apostol “Introduction to Analytic Number Theory", tentatively Chapters 1–4, 6, 7, 11–13. The book should be purchased through an Internet bookstore.

If you already took MA3001, you may alternatively register for TMA4310 (or vice versa, if you already took TMA4310).

Teaching Hours

• Wednesday 12:15 - 14:00, room 734, Sentralbygg 2
• Thursday 08:15 - 10:00, room 734, Sentralbygg 2.

The first lecture will be on Thursday January 9.

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. In particular, the exercises for March 13 and March 27 are essential for understanding basic points such as von Mangoldt's formula and the link between $\psi(x)-x$ and $\pi(x)-\text{Li}(x)$.

Contents of the lectures

• Lecture 1, January 9: Section 1.6 (elementary proof of the divergence of the series of reciprocals of the primes) and Sections 2.1 - 2.5 (the Möbius function $\mu(n)$, Euler's totient function $\varphi(n)$; basic properties and the relation between these two functions).
• Lecture 2, January 15: Sections 2.6-2.7 (Dirichlet multiplication/convolution), Section 2.8 (the von Mangoldt function $\Lambda(n)$), Sections 2.9-2.11 (multiplicative functions).
• Lecture 3, January 22: Sections 3.1-3.5 (Big oh notation, Euler summation, some asymptotic formulas, average order of $d(n)$). See Jeffrey Lagarias's paper Euler's constant: Euler's work and modern developments for an interesting survey; this paper appeared in Bull. Amer. Math. Soc. 50 (2013), 527–628.
• Lecture 4, January 23: Sections 3.6 (average order of $\sigma_{\alpha}(x)$), 3.7 (average order of $\varphi(n)$), 4.1-4.3 (the functions $\pi(x)$, $\vartheta(x)$, $\psi(x)$, $\Lambda(x)$ and relations between them). In this lecture we are concerned with the average order of $\sigma(x)$, but be aware that finding the best upper bound is of considerable interest; see Jeffrey Lagarias's paper An elementary problem equivalent to the Riemann hypothesis, which appeared in Amer. Math. Monthly 109 (2002), 534–543.
• Lecture 5, January 29: Sections 4.4-4.5 (equivalent forms of the prime number theorem and related inequalities) and 4.8 (the asymptotic behavior of $\sum_{p\le x} p^{-1}$; we will at this stage be less precise than in Apostol's book, the precise asymptotics will be considered later).
• Lecture 6 - 7, February 5 - 6 (by Winston Heap): Dirichlet characters and primes in arithmetic progressions
• No lecture on February 12 since Dirichlet's theorem has already been covered through Winston Heap's lectures. Those interested can come to work on the problems for Chapters 6 and 7, and I will be there to help you with this.
• Lecture 8, February 19: 11.1-11.5 (Generalities about Dirichlet series, multiplication of Dirichlet series and Dirichlet convolution, Euler products).
• Lecture 9, February 20: 11.6-7, 11.9 (More about Dirichlet series; analytic properties, convergence).
• Lecture 10, February 26: 11.12 (Perron's formula + the Mellin transformation and the first step in the proof of von Mangoldt's explicit formula for Chebyshev's function $\psi(x)$ (not treated in Apostol, but can be found e.g. in Chapter 17 of H. Davenport's "Multiplicative Number Theory").
• Lecture 11, March 5: 12.2 and beyond (basic properties of the Gamma function including the functional equation and product representation; you can find a presentation of this material here).
• Lecture 12, March 6: Riemann's proof of the functional equation for $\zeta(s)$ via the Poisson summation formula. (See e.g. Chapter 8 of H. Davenport's "Multiplicative Number Theory".)
• Lecture 13, March 12 (by Winston Heap): Entire functions of order 1 and product representation of $\xi(s)$. (NB! We will not treat functional equations for general L-functions and the Hurwitz zeta function as in Chapter 12 in Apostol, but instead focus only on $\zeta(s)$ and discuss in more depth analytic properties of this function. The material covered in this lecture can be found in Chapters 11 and 12 of H. Davenport's "Multiplicative Number Theory".) Lecture Notes
• Lecture 14, March 19: Von Mangoldt's formula (see Chapter 17 of Davenport), including a discussion of the problems from March 13.
• Lecture 15, March 20: Additional discussion of the problems from March 13.
• Lecture 16, March 26: Zero-free regions for $\zeta(s)$ (see Apostol 13.5, 13.8, and Chapter 13 in Davenport) and proof of the prime number theorem from von Mangoldt's formula (see Chapter 18 in Davenport).
• Lecture 17, April 2: (final lecture): Final remarks on the proof of the prime number theorem; the error term, zeros of the zeta function, and the proof in Chapter 13 of Apostol.

Exercises

• Exercises for January 16: 7, 8, 11, 12, 30 from Chapter 1; 1, 2, 6, 14, 18, 19, 21, 26 from Chapter 2. Solutions to some of the problems. (You may also find proposed solutions on the Internet, but please be aware that not all of them are correct.)
• Exercises for January 30: 1, 2, 3, 11, 18 from Chapter 3; 5, 7, 9, 11, 15, 18, 19 from Chapter 4. Solutions to some of the problems; Solution to Problem 4.9.
• Exercises for February 13: 14, 15, 16, 18 from Chapter 6; 1, 2, 3, 4, 6, 8 from Chapter 7. Solutions
• Exercises for February 27: 1a, 1b, 1c, 1d, 2a, 2c, 2d, 3, 4, 6, 12 from Chapter 11. Solutions
• Exercises for March 13 (NB! In an earlier version, there was a mistake in the formulation of Problem 4. This has now been corrected.) Solutions
• Exercises for March 27: 3, 4 from Chapter 13 in Apostol plus these problems. Solutions

Note on Arithmetical Functions

Tjerand Sildre sent me this file, where some basic arithmetical functions have been programmed in Python. He wanted to share it with all of you in case you might find it interesting and helpful.

Oral presentations

During the last three lectures, the students should give short presentations of topics assigned to them. These are topics that are not covered by the lectures. You may choose a topic yourself (to be approved by me) or choose one from the following list:

1. Euler's work on his constant (based on Lagarias's paper) (Brage Sæth)
2. Bernoulli numbers and $\zeta(2n))$ (Marte Oldervoll)
3. Mertens's theorems and Mertens's constant (Therese Mardal Hagland)
4. The Bertrand–Chebyshev theorem, including Ramanujan and Erdős's work on it (Emilie Arentz-Hansen)
5. Ramanujan primes (Reidun Persdatter Ødegaard)
6. General distribution of nontrivial zeros of $\zeta(s)$ (Jürgen Steininger)
7. Zeros on the critical line, including density results (Øistein Søvik)
8. Zero-free regions for $\zeta(s)$ (Thor Mikkel Nordahl)
9. The error term in the prime number theorem and zero-free regions (Ola Jermstad)
10. The Lindelöf hypothesis (Gunnar Sveinsson)
11. Mean value theorems - results and conjectures
12. Dirichlet L-functions for which RH fails (see e.g. Titchmarsh)
13. Dirichlet's divisor problem, including Voronoi's summation formula (Petter Nyland)
14. Elementary sieve methods and Brun's theorem on twin primes (Tjerand Silde)
15. Voronin's universality theorem (see e.g. Titchmarsh) (Håkon Bølviken)
16. Lagarias's version of Guy Robin's criterion (see Lagarias's paper) (Eirik Mork)
17. The Beurling–Nyman condition for RH (Sofia Lindqvist)
18. Li's criterion for RH (Håvard Bakke Bjerkevik)
19. The Bohr–Cahen formulas for abscissas of convergence and the growth of $\sum_{n\le x} \mu(n)$ (Espen Danielsen).

NB! Students choosing topics from the subgroups 1-3, 4-5, 6-9, 10-11 should communicate with each other when preparing their presentations, since the topics in the respective subgroups are closely related. Please make your choice by email to me before March 1. Topics will be assigned according to the "first-come, first-served" principle. I will put your name into the list above as soon as a topic has been assigned to you.

The aim of the presentations is to convey to your peers what the topic is about and the most important and interesting problems and results associated with it. You are not expected to study proofs of deep theorems, but you should be able to say a little more than what we can find on the Wikipedia.

Schedule

Thursday April 3

• 08:15 - 08:30 Eirik Mork: Lagarias's version of Guy Robin's criterion
• 08:35 - 08:50 Sofia Lindqvist: The Beurling–Nyman condition for RH
• 08:55 - 09:10 Håvard Bakke Bjerkevik: Li's criterion for RH
• 09:20 - 09:35 Espen Danielsen The Bohr–Cahen formulas for abscissas of convergence and the growth of $\sum_{n\le x} \mu(n)$
• 09:40 - 09:55 Håkon Bølviken: Voronin's universality theorem

Wednesday April 9

• 12:10 - 12:25 Brage Sæth: Euler's work on his constant
• 12:30 - 12:45 Marte Oldervoll: Bernoulli numbers and $\zeta(2n)$
• 12:50 - 13:05 Therese Mardal Hagland: Mertens's theorems and Mertens's constant
• 13:10 - 13:25 Emilie Arentz-Hansen: The Bertrand–Chebyshev theorem, including Ramanujan and Erdős's work on it
• 13:30 - 13:45 Reidun Persdatter Ødegaard: Ramanujan primes
• 13:50 - 14:05 Jürgen Steininger: General distribution of nontrivial zeros of $\zeta(s)$

Thursday April 10

• 08:10 - 08:25 Øistein Søvik: Zeros on the critical line, including density results
• 08:30 - 08:45 Thor Mikkel Nordahl: Zero free regions for $\zeta(s)$
• 08:50 - 09:05 Ola Jermstad: The error term in the prime number theorem and zero-free regions
• 09:10 - 09:25 Gunnar Sveinsson: The Lindelöf hypothesis
• 09:30 - 09:45 Petter Nyland: Dirichlet's divisor problem, including Voronoi's summation formula
• 09:50 - 10:05 Tjerand Silde: Elementary sieve methods and Brun's theorem on twin primes Presentation

Examination

In addition to the oral presentation, there will be an oral examination that will take place on April 30 and May 5. You should be prepared to answer questions about topics from the syllabus. The examination may take up to 25 minutes.

The final grade will depend on the oral presentation (counts 50 %) and the oral examination (counts 50 %).

Schedule for the examination

Wednesday April 30 Room 922 Sentralbygg 2

• 09:00 Marte Oldervoll
• 09:30 Øistein Søvik
• 10:00 Therese Mardal Hagland
• 10:30 Reidun Persdatter Ødegaard
• 11:00 Tjerand Silde
• 11:30 Brage Sæth
• 13:00 Eirik Mork
• 13:30 Thor Mikkel Nordahl
• 14:00 Ola Jermstad

Monday May 5 Room 922 Sentralbygg 2

• 09:30 Petter Nyland
• 10:00 Sofia Lindqvist
• 10:30 Jürgen Steininger
• 11:00 Håkon Bølviken
• 11:30 Gunnar Sveinsson
• 13:00 Håvard Bakke Bjerkevik
• 13:30 Espen Danielsen