MA3150 Analytic Number Theory, Spring 2019

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, 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.

Kristian Seip: Office 956 in SB II, kristian [dot] seip [at] ntnu [dot] no

• Monday 8:15 - 10:00, 656 SB2
• Tuesday 12:15 - 14:00, GL-VA VA2

The first lecture: January 8. It was decided during the first lecture that the lecture originally scheduled for Wednesday be shifted to Monday. There is therefore NO LECTURE on January 9.

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.

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.

The final lecture took place on March 19. You are supposed to work on the topic for your oral presentation during the four remaining weeks of the semester.

• Lecture 1, January 8: The Poisson summation formula with the example of the Gaussian function. Definition of 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; se also Ch. Vanden Eynden, "Proofs that \(\sum 1/p\) diverges", Amer. Math. Monthly 87 (1980), 394–397). See this note for a more precise consequence of Euler's argument.
• Lecture 2, January 14: Apostol, sections 2.1 - 2.9. 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 3, January 15: Apostol, sections 2.10 - 2.12, 3.1 - 3.4 (see also Thm. 4.2 in 4.3 which implies (6) on page 54). More on multiplicative functions, big oh notation, Abel summation and the Euler–Maclaurin summation formula, some asymptotic formulas, meromorphic continuation of \( \zeta(s) \). (See this note for both Abel summation and the Euler–Maclaurin formula applied to \( \zeta(s) \).)
• Lecture 4, January 21: Apostol, sections 3.5, 3.7, 3.11, 4.1 - 4.3. Further applications of Abel summation and the Euler–Maclaurin formula (more on the analytic continuation of \( \zeta(s) \), relation between \( \pi(x) \) and \( \psi(x) \), a weak version of Stirling's formula), average order of \(d(n)\); for the average order of \(\varphi(n)\), see Exercise 1, problems 3.4, 3.5 in Apostol.
• Lecture 5, January 22: Apostol 3.10 - 3.11, 4.5 - 4.8; see also Ch. 7 in Davenport. Chebyshev's bounds on \( \pi(x) \) and Mertens's theorem on the asymptotics of \(\sum_{p\le x} p^{-1}\) (see this note for a short route to these results; see also here for an interesting account of Mertens's theorems); the sum \( \sum_{n\le x} \mu(n)/n \); the number of square-free numbers less than or equal to \( x \) (see Ex. 2.6 in Apostol for the "groundwork").
• Special lecture related to this course: Christian Skau, "A century of Brun's sieve - a momentous event in prime number history", January 25, 12:15 - 13:00, 1329 SB2. Coffee and cakes are served during the lecture.
• Lecture 6, January 28: Preparation for our study of \(\zeta(s)\): the basics of the Gamma function; functional equation, infinite product representation, reflection formula, Stirling's formula. See Note on the Gamma function.
• Lecture 7, January 29: The gamma function continued; proof of Stirling's formula and some consequences, Legendre's duplication formula. Riemann's memoir and the functional equation for \(\zeta(s)\) (see Ch. 8 in Davenport).
• Lecture 8, February 4: Entire functions of order 1 and product representation of ξ(s) (see Ch. 11 and 12 in Davenport). (Notice our usage of Cauchy estimates and Jensen's formula from complex analysis.)
• Lecture 9, February 5: We have now two representations of ζ′(s)/ζ(s): 1) in terms of \( \Lambda(n) \) via the Euler product of ζ(s) and 2) in terms of the pole and the zeros of ζ(s). In this and the next lectures, we will see what this relation leads to. We begin by deducing the facts that \( N(T+1)-N(T)=O(\log T) \) and that \( \zeta'(s)/\zeta(s)\) for large \( t \) is \( \sum_{|\gamma-t|\le 1} \frac{1}{s-\rho} + O(\log |t|) \) (see Ch. 15 of Davenport).
• Lecture 10, February 11: We establish the argument principle and apply it to deduce the Riemann–von Mangoldt formula for \( N(T) \). We introduce and discuss the function \( S(T) \), which can be viewed as the argument of \( \zeta(s) \) on the critical line. We show that \( S(T)=O(\log T) \) (see Ch. 15 of Davenport). (The lecture is given by Kamalakshya Mahatab.)
• Lecture 11, February 12: We discuss the most interesting points of Exercise 1 and Exercise 2. (The lecture is given by Kamalakshya Mahatab.)
• NB! The lecture on February 18 is cancelled because of sickness absence of the lecturer.
• Lecture 12, February 19: We establish Perron's formula (this is done in a special case in Ch. 17 of Davenport and in the general case in 11.12 of Apostol).
• Lecture 13, February 22, 14:15-16, Room 734 SB2 (This is an extra lecture): We now 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 explicit formula for the Chebyshev function \(\psi(x)\) (see Ch. 17 in Davenport). We finish the lecture by observing what the Riemann hypothesis implies about the prime counting function \( \pi(x) \).
• Lecture 14, February 25: We establish the zero-free region of de la Vallée Poussin (Ch. 13 of Davenport) and use it to prove the prime number theorem (Ch. 18 of Davenport).
• Lecture 15, February 26: We start preparing for Dirichlet's theorem on primes in arithmetic progressions by considering some facts about finite Abelian groups (see Ch. 6 in Apostol and Ch. 1 in Davenport), including the character group and orthogonality of the characters. Finally, we define Dirichlet characters and consider the simplest examples (for \( k \le 5\)).
• Lecture 16, March 4. We start discussing Dirichlet's theorem. We will mostly follow Apostol's proof (see Ch. 7 of Apostol), but will use some arguments from Davenport's Ch. 4 as well. We may interpret "Dirichlet's theorem" either as the easier "Euler formula" \( \sum_{p\in A(h,k)} p^{-\sigma}=\frac{1}{\phi(k)}\log\frac{1}{\sigma-1}+O(1), \sigma>1\) or 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 17, March 5: We finish the proof of Dirichlet's theorem, stressing that the key point is to show that \( L(1,\chi)\neq 0 \) for all nonprincipal characters \( \chi \).
• Lecture 18, March 11. We will in the remaining lectures discuss the ideas that eventually lead to the prime number theorem for arithmetic progressions (see Ch. 20 and 22 in Davenport). We begin by introducing the notion of a primitive character, see Ch. 5 of Davenport and Ch. 8 of Apostol. We will mostly follow Apostol, relying on finite Fourier transforms (Gauss sums). This lecture will roughly cover 8.1 - 8.2, 8.5 - 8.7 in Apostol. (Be aware that Theorem 8.11 in Apostol is just Parseval's relation for the Fourier transform on \( \mathbb{Z}_k \).)
• Lecture 19, March 12. We continue our discussion of primitive characters and Gauss sums, covering roughly 8.8 - 8.12 in Apostol.
• Lecture 20, March 18. We establish the functional equation for \( L(s,\chi) \) when \( \chi \) is a primitive character, following Ch. 9 in Davenport.
• Lecture 21, March 19 (FINAL LECTURE). We recall our route to the prime number theorem and discuss how to proceed to obtain the corresponding result for arithmetic progressions. The lecture will include some words on zero-free regions and Siegel zeros (see Ch. 14 in Davenport).

You are welcome to work on the exercises in room 1329 SB2 on Friday 14:00–15:00, under my guidance, as indicated below. Solutions to the problems will be provided in due course.

• Exercise 1, from Apostol: 2.1, 2.2, 2.6, 2.21, 2.26, 3.1, 3.2, 3.4, 3.5, 4.7, 4.18, 4.19 (a) (guidance offered on January 18 and 25).
• Exercise 2 (guidance offered on February 1 and February 8).
• Exercise 3, from Apostol: 11.1 (a)–(d), 11.3, 13.2 (observing that \( A(x)=\pi(x)+\pi(x^{1/2})/2+\pi(x^{1/3})/3+\cdots \), you may prove the stronger result that \( A(x)=\pi(x)+O(\sqrt{x})\)), 13.3, 13.4 (guidance offered on March 1).
• Exercise 4, from Apostol: 6.14, 6.15, 7.1, 7.2, 7.3, 7.4, 7.6 plus this problem that was left as a challenge in the lecture on March 5 (guidance offered on March 8 and March 15).
• Exercise 5, from Apostol: 8.5, 8.6, 8.7, 8.8, 8.9 (guidance offered on March 15 and March 22).

As part of the oral exam, 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. Mertens's theorems and Mertens's constant
2. The Bertrand–Chebyshev theorem, including Ramanujan and Erdős's work on it
3. Ramanujan primes (chosen by Oskar Vikhamar-Sandberg)
4. Skewes's number and sign changes in \( \pi(x)-\operatorname{li}(x) \)
5. General distribution of nontrivial zeros of \(\zeta(s)\)
6. Zeros on the critical line, including density results
7. The error term in the prime number theorem and zero-free regions
8. The Lindelöf hypothesis and the density hypothesis
9. Mean value theorems - results and conjectures
10. Zeta functions for which RH fails
11. Dirichlet's divisor problem, including Voronoi's summation formula
12. Elementary sieve methods and Brun's theorem on twin primes
13. Voronin's universality theorem and value distribution of the Riemann zeta function
14. Lagarias's version of Guy Robin's criterion
15. The Beurling–Nyman condition for RH
16. Li's criterion for RH
17. The Bohr–Cahen formulas for abscissas of convergence and the growth of \(\sum_{n\le x} \mu(n)\).
18. 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)
19. Approximations of \(\zeta(s)\), including the approximate functional equation
20. The Riemann–Weil explicit formula
21. Siegel zeros.

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. You may choose to give a blackboard lecture or a Beamer/Powerpoint presentation. Each presentation should last for about 15–20 minutes.

Please let me know your choice of topic before March 29. There should be only one student per topic. Once a topic is chosen, I will make a note of it in the list above. Accordingly, we follow the principle of "first-come, first-served" when assigning topics.

The oral presentations will be given on May 8. You are strongly encouraged to be present at all the presentations! Oral examinations will take place on May 9. Both events will take place in Room 656 SB2.

Before the Easter break, I will be available for consultation until April 3. I will be traveling April 4 – 11, and will again be available on April 12. After the Easter break, I will be available only May 6 – 7.

You may in principle come at "any" time during the days I am in my office, but I would recommend that you contact me in advance to make an appointment.