# Analytisk tallteori våren 2017

## (MA3001 Mastergradsseminar i matematikk/TMA4310 Matematiske emner, videregående kurs)

In the spring semester of 2017, the topic of MA3001 will be Analytic Number Theory. If you already took MA3001, you may alternatively register for TMA4310.

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 localization of the zeros of this meromorphic function is closely related 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, 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.

## Lectures

The first lecture was on Friday January 13, but we have agreed on a new schedule beginning on January 16 (NB! Notice that all lectures are in room 656):

- Monday 08:15 - 10:00, room 656, Sentralbygg 2
- Thursday 16:15 - 18:00, room 656, Sentralbygg 2.

## Textbook

Harold Davenport, “Multiplicative Number Theory", Third Edition, Springer Graduate Texts in Mathematics. The lectures will to a large extent 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.

## Contents of the lectures

- Lecture 1, January 13: Definition of the Riemann zeta function, Euler's and Erdős's proof of the divergence of the series of reciprocals of the primes (from Davenport Ch. 1 and Ch. Vanden Eynden, "Proofs that \(\sum 1/p\) diverges", Amer. Math. Monthly
**87**(1980), 394–397), the Möbius function \(\mu(n)\), Euler's totient function \(\varphi(n)\); basic properties and the relation between these two functions (see sections 2.1 - 2.5 in Apostol). - Lecture 2, January 16: More on Dirichlet convolution, Möbius inversion, the von Mangoldt function \(\Lambda(n) \), multiplicative functions (see sections 2.6–2.11 in Apostol).
- Lecture 3, January 19: Big oh notation, Abel summation and Euler's summation formula, some asymptotic formulas, average order of \(d(n)\)) and \(\varphi(n)\) (see sections 3.1-3.5, 3.7, 4.1 in Apostol).
- Lecture 4, January 23: Review of our work on summatory functions; more applications of Abel summation and Euler's summation formula; a weak version of Stirling's formula; the Mertens function (partial sums of the Möbius function); Mertens's theorem on the asymptotics of \(\sum_{p\le x} p^{-1}\) (see sections 3.9-3.11, 4.6-4.8 in Apostol and Ch. 7 in Davenport; see also here for an interesting account of Mertens's theorems).
- Lecture 5, January 30: Brief review of last week's exercises. Preparation for our study of \(\zeta(s)\): the basics of the Gamma function. See Note on the Gamma function.
- Lecture 6, February 2: Summary of our work on \( \Gamma(s)\), the Poisson summation formula, Riemann's memoir and the functional equation for \(\zeta(s)\) (see Ch. 8 in Davenport).
- Lecture 7, February 6: Proof of the functional equation for \( \zeta(s) \) (see ch. 8 in Davenport); entire functions of order 1 and product representation of \(\xi(s)\) (see Ch. 11 and 12 in Davenport). (Notice our usage of Cauchy estimates and Jensen's formula from complex analysis.
- Lecture 8, February 13: We have now two representations of \( \zeta'(s)/\zeta(s)\): 1) in terms of von Mangoldt's function via the Euler product of \(\zeta(s) \) and 2) in terms of the pole and the zeros of \(\zeta(s)\). In the next lectures, we will see what this relation leads to. We begin by studying the general distribution of the zeros of \(\zeta(s)\) (see Ch. 15 of Davenport). Here we will use the argument principle from complex analysis.
- Lecture 9, February 16: We finish our study of \(N(T)\) (number of nontrivial zeros of \(\zeta(s)\) up to height \( T\)) by estimating \(S(T)\) via a suitable estimate of \(\zeta'(s)/\zeta(s)\) (see Ch. 15 of Davenport). We then proceed to discuss Perron's formula (this is done in a special case in Ch. 16 of Davenport and in the general case in 11.12 of Apostol).
- Lecture 10, February 20: We apply Perron's formula with remainder term, our estimates for \(\zeta'(s)/\zeta(s)\), and our knowledge about the nontrivial zeros of \( \zeta(s)\) to deduce von Mangoldt's effective formula for the Chebyshev function \(\psi(x)\) (see Ch. 17 in Davenport).
- Lecture 11, February 27: We prove and discuss the prime number theorem (a combination of Ch. 13, 17, 18 in Davenport). Notice also the relation to the exercises from February 23.
- Lecture 12, March 2: Concluding remarks about the prime number theorem and zero-free regions. We then start our preparation for Dirichlet's theorem on primes in arithmetic progressions by considering some facts about finite Abelien groups (see Ch. 6 in Apostol and Ch. 1 in Davenport)
- Lecture 13, March 6: We continue our preparation for the proof of Dirchlet's theorem on primes in arithmetic progressions. We introduce the Dirichlet characters (see end of Ch. 6 in Apostol) before embarking on the proof of the theorem. We will mostly follow Apostol's proof (see Ch. 7 of Apostol).
- NB! NO LECTURE March 9 because there is a guest lecture organized by Delta at the same time.
- Lecture 14, March 16: We continue our discussion of the proof of Dirichlet's theorem in terms of the Mertens formula \( \sum_{p\le x, p\in A(h,k)}\frac{\log p}{p}=\frac{1}{\phi(k)}\log x+O(1) \), where \(A(h,k)\) is the arithmetic progression \(\{m: m=h+nk, n\ge 0\}\) and \((h,k)=1\).
- Lecture 15, March 20: We finish the proof of Dirichlet's theorem by showing that \(L(1,\chi)\neq 0 \) whenever \( \chi\) is a nonprincipal character. Concluding remarks about Dirichlet's theorem.
- Lecture 16, March 23: We will in the remaining lectures briefly sketch some of the basic ideas that eventually lead to the Siegel–Walfisz theorem (Ch. 22 in Davenport). In the present lecture, we will define Dirichlet \(L\)-functions, discuss some of their basic analytic properties, and present Davenport's proof of Dirichlet's theorem (see pp. 31–34 in Davenport). We will during the lecture also mention a few highlights from Ch. 11 in Apostol pertaining to the functions \(L(s,\chi)\).
- Lecture 17, March 27: Primitive characters (parts of Ch. 8 in Apostol and Ch. 5 in Davenport); we discuss among other things the representation of general nonprincipal characters in terms of primitive characters and the corresponding relation between their \(L\) functions.
- Lecture 18, March 30 (FINAL LECTURE): This lecture will be about Gauss sums and in particular Gauss sums for primitive characters (see 8.5 – 8.7 in Apostol); we will then finish with an indication of how to proceed to obtain the functional equation for Dirichlet \(L\)-functions for primitive characters (see Ch. 9 in Davenport).

## Exercises

- January 26: From Apostol: 2.1, 2.2, 2.6, 2.21, 3.1, 3.2, 3.4, 3.5, 4.5, 4.7, 4.18, 4.19.
- February 23: From Apostol: 13.3, 13.4 plus these problems.
- March 11: From Apostol: 6.14, 6.15, 6.16, 7.1, 7.2, 7.3, 7.4, 7.6.

## Oral presentations

During the lectures on April 27 and May 4, 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.

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:

- Euler's work on his constant (based on Lagarias's paper) (
`Magnus Ringerud`

) - Bernoulli numbers and \(\zeta(2n)\) (
`Eivind Otto Hjelle`

) - Mertens's theorems and Mertens's constant (
`Amirhossein Kazemi`

) - The Bertrand–Chebyshev theorem, including Ramanujan and Erdős's work on it (
`Didrik Fosse`

) - Ramanujan primes (
`Mona-Lena Norheim`

) - General distribution of nontrivial zeros of \(\zeta(s)\)
- Zeros on the critical line, including density results
- Zero-free regions for \(\zeta(s)\)
- The error term in the prime number theorem and zero-free regions
- The Lindelöf hypothesis (
`Cecilie Bjørnsdotter Raustein`

) - Mean value theorems - results and conjectures
- Zeta functions for which RH fails (see e.g. Titchmarsh) (
`Erlend Due Børve`

) - Dirichlet's divisor problem, including Voronoi's summation formula (
`Nigus Girmay Teklehaymanot`

) - Elementary sieve methods and Brun's theorem on twin primes (
`Peter Flydal`

) - Voronin's universality theorem (see e.g. Titchmarsh)
- Lagarias's version of Guy Robin's criterion (see Lagarias's paper)
- The Beurling–Nyman condition for RH
- Li's criterion for RH (
`Christopher Kvarme`

) - The Bohr–Cahen formulas for abscissas of convergence and the growth of \(\sum_{n\le x} \mu(n)\).
- Alternate proofs of the functional equation for \( \zeta(s)\) (Titchmarsh gives 7 proofs; take a look and make your own selection of some of them)
- Approximations of \(\zeta(s)\), including the approximate functional equation.

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.

Each presentation should last for about 15–20 minutes.

### Schedule (NB! New Version as of April 24)

#### Thursday May 4

- 13:00 - 13:20
`Magnus Ringerud`

: Euler's work on his constant - 13:20 - 13:40
`Eivind Otto Hjelle`

: Bernoulli numbers and \(\zeta(2n)\) - 13:45 - 14:05
`Amirhossein Kazemi`

: Mertens's theorems and Mertens's constant - 14:05 - 14:25
`Didrik Fosse`

: The Bertrand–Chebyshev theorem, including Ramanujan and Erdős's work on it - 14:30 - 14:50
`Mona-Lena Norheim`

: Ramanujan primes

- 15:00 - 15:20
`Cecilie Bjørnsdotter Raustein`

: The Lindelöf hypothesis - 15:20 - 15:40
`Erlend Due Børve`

: Zeta functions for which RH fails - 15:45 - 16:05
`Nigus Girmay Teklehaymanot`

: Dirichlet's divisor problem - 16:05 - 16:25
`Peter Flydal`

: Elementary sieve methods and Brun's theorem on twin primes - 16:30 - 16:50
`Christopher Kvarme`

: Li's criterion for RH

## Examination

### Schedule for the examination

#### Friday May 12 Room 656 Sentralbygg 2

- 09:00
`Magnus Ringerud`

- 09:30
`Eivind Otto Hjelle`

- 10:00
`Amirhossein Kazemi`

- 10:30
`Didrik Fosse`

- 11:00
`Mona-Lena Norheim`

: - 11:30
`Cecilie Bjørnsdotter Raustein`

- 13:00
`Erlend Due Børve`

- 13:30
`Nigus Girmay Teklehaymanot`

- 14:00
`Peter Flydal`

- 14:30
`Christopher Kvarme`

## Referansegruppe

- Final meeting will take place on April 24.

## Analytic Number Theory 2014

See Analytisk tallteori våren 2014 for the previous version of this course.