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 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.

Kristian Seip: Office 956 in SB II, kristian.seip@ntnu.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.

Tor Kringeland

Lars Magnus Øverlier

• Minutes from the first meeting January 14.

• 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. See Note on the Gamma function.
• Lecture 7, January 29: Riemann's memoir and the functional equation for \(\zeta(s)\) (see Ch. 8 in Davenport). If time permits, we will then move on to Ch. 11 in Davenport, to arrive at the canonical factorization of \( \zeta(s) \) in terms of its pole and zeros.

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).

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
4. General distribution of nontrivial zeros of \(\zeta(s)\)
5. Zeros on the critical line, including density results (chosen by Terje Bull Karlsen)
6. The error term in the prime number theorem and zero-free regions
7. The Lindelöf hypothesis and the density hypothesis
8. Mean value theorems - results and conjectures
9. Zeta functions for which RH fails
10. Dirichlet's divisor problem, including Voronoi's summation formula
11. Elementary sieve methods and Brun's theorem on twin primes
12. Voronin's universality theorem and value distribution of the Riemann zeta function
13. Lagarias's version of Guy Robin's criterion
14. The Beurling–Nyman condition for RH
15. Li's criterion for RH
16. The Bohr–Cahen formulas for abscissas of convergence and the growth of \(\sum_{n\le x} \mu(n)\).
17. 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)
18. Approximations of \(\zeta(s)\), including the approximate functional equation
19. The Riemann–Weil explicit formula.

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.