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 and Fourier Analysis corresponding to what is covered by MA2106 or TMA4120 Matematikk 4K.

Kristian Seip
Office: Room 956, Sentralbygg 2
Email: kristian [dot] seip [at] ntnu [dot] no

Office hours (from March 16):

• Tuesday 14–16
• Thursday 08–10.

• Tuesday 15:15–17:00, D4-132 Realfagbygget
• Thursday 08:15–10:00, Kjel 24

The first lecture will take place on January 10.

• Lecture 1, January 10: A short historical introduction. 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 12: Apostol, sections 2.1 - 2.7, 2.9 -2.10: The von Mangoldt function \(\Lambda(n)\), the Möbius function \(\mu(n)\), Euler's totient function \(\varphi(n)\); basic properties and the relation between these two functions, Dirichlet convolution, Möbius inversion, multiplicative and completely multiplicative functions.
• Lecture 3, January 17: Apostol, sections 2.8, 2.11, 3.1 - 3.4 (see also Thm. 4.2 in 4.3 which implies (6) on page 54). Big oh notation, Abel summation and the Euler summation formula, some asymptotic formulas, meromorphic continuation of \( \zeta(s) \). (See this note for both Abel summation and the Euler formula applied to \( \zeta(s) \).)
• Lecture 4, January 19: Apostol, sections 3.5, 3.11, 4.5, 4.8 (see also Ch. 7 in Davenport). More on sums of arithmetic functions; Dirichlet's divisor problem, a weak version of Stirling's approximation; Chebyshev's bounds on \( \pi(x) \); 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).
• Lecture 5, January 24: We finish our discussion of Mertens's theorem. We prepare for our study of \(\zeta(s)\) by looking at the basics of the Gamma function (which is ubiquitous in analytic number theory); functional equation, Gauss's product formula and infinite product representation. See Winston Heap's note on the Gamma function. What we actually need, is summarized in Ch. 10 of Davenport.
• Lecture 6, January 26: We continue our study of the Gamma function; we deduce Euler's reflection formula, Legendre's duplication formula, and Stirling's formula which yields precise information about gamma off the negative real half-line. (See Winston Heap's notes for the details that were not treated during the lecture.)
• Lecture 7, January 31: As a final preparation for our study of \(\zeta(s)\), we deduce the Poisson summation formula from Fourier analysis, and we compute the Fourier transform of the Gaussian. We then turn to Riemann's memoir and the functional equation for \(\zeta(s)\) (see Ch. 8 in Davenport). We try to explain why it is natural to use Poisson summation when we establish the functional equation.
• Lecture 8, February 2: Entire functions of order 1 and product representation of \(\xi(s)\) (see Ch. 11 and 12 in Davenport). We take a somewhat different route to this product representation via the crude bound \( N(T+1)-N(T)=O(\log T) \), which we deduce from Jensen's formula. Note that Jensen's formula gives a precise relation between the growth of analytic functions and the distribution of their zeros.
• Lecture 9, February 7: We finish our deduction of the product representation of \(\xi(s)\) over the nontrivial zeros of \(\zeta(s)\). Before we start our preparation for the proof of the prime number theorem, we establish the Riemann–von Mangoldt formula for \(N(T)\) which we get from the functional equation, the argument principle, and Stirling's formula.
• Lecture 10, February 9: We finish our discussion of the Riemann–von Mangoldt formula by establishing the classical bound \( S(T)=O(\log T) \). This we get from the fact that, in the critical strip, \( \zeta'(s)/\zeta(s)\) for large \( t \) is \( \sum_{|\gamma-t|\le 1} \frac{1}{s-\rho} + O(\log |t|) \) (see Ch. 15 of Davenport), an estimate that will also be needed in our deduction of von Mangoldt's explicit formula for \(\psi(x)\). If time permits, we will start our work on that formula by establishing 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 11, February 14: We finish the proof of Perron's formula with remainder term. We then apply it to express Chebyshev's function \(\psi(x)\) as a complex line integral plus a controllable remainder term, depending on the length of the vertical line segment over which we integrate. The next step is then to apply the residue theorem to find another expression for this complex line integral that involves the nontrivial zeros of \(\zeta(s)\). This will eventually lead us to von Mangoldt's explicit formula for \(\psi(x)\) (see Ch. 17 in Davenport). We finish the lecture by observing what the Riemann hypothesis implies about the prime counting function \( \pi(x) \).
• Lecture 12, February 16: We start by proving that the supremum of the real parts of the nontrivial zeros equals the infimum of the \(\theta>0\) such that we have \(\psi(x)=x+O(x^{\Theta})\). We then establish the zero-free region of de la Vallée Poussin (Ch. 13 of Davenport) and see how this leads, via von Mangoldt's explicit formula, to a proof the prime number theorem (Ch. 18 of Davenport).
• Lecture 13, February 21: We summarize a bit what we have done so far and discuss exercises 1 and 2.
• Lecture 14, February 23: 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 the \(\phi(k)\) Dirichlet characters modulo \(k\). (Apology: Inadvertently, I did not record the first half of the lecture. But the most important points are: The construction of the character group and the orthogonality relations of the characters.)
• Lecture 15, February 28: We recapture the definition of Dirichlet characters and their basic properties. We look at the simplest examples (for \( k \le 5\)) and note that Legendre symbols constitute an interesting family of real Dirichlet characters. (You may also notice that Davenport in Ch. 1 presents an explicit construction of the Dirichlet characters to a prime modulus.) We prepare for the proof of Dirichlet's theorem by looking at Dirichlet \(L\)-functions \(L(\chi,s)\) and some of their basic properties. Here we mostly follow Davenport's Ch. 4.
• Lecture 16, March 2: We prove that \(L(1,\chi)\neq 0\) also when \(\chi\) is a real nonprincipal character. Here we use de la Vallée Poussin's argument (see Ch. 4 of Davenport). We have thus finished the proof of Dirichlet's theorem, with the Mertens-type bound \(\sum_{p\le x, p\equiv a (\operatorname{mod} k)}1/p=\tfrac{1}{\phi(k)}\log\log x+O(\log\log\log x)\). We then take a look at how this can be improved via the Mertens theorem we already have, namely \(\sum_{p\le x} \tfrac{\log p}{p}=x+O(1).\) Here we partly follow Apostol 7.3–6.
• Lecture 17, March 7: We finish the "Mertens" version of Dirichlet's theorem. Note that this should be considered an elementary result, although we keep our complex analytic intuition in mind when relating \(\sum_{n\le x} \chi(n)\Lambda(n) n^{-1} \) to \(\sum_{n\le x} \mu(n) \chi(n) n^{-1}\). What remains is to show that the latter partial sums are bounded. NB! For the next couple of weeks, we will mainly summarize what we have covered during the preceding lectures. We begin today by going slowly through the proof of the bound \(N(T+1)-N(T)=O(\log T)\).
• Lecture 18, March 9: Our proof of PNT is quite elaborate, so it may be worthwhile taking another look at the various steps and how they fit together. This lecture aims at doing exactly this.
• Lecture 19, March 14: We discuss again the importance of zero-free regions and take another look at that found by de la Vallée Poussin. This was the last ordinary lecture.
• On March 30, we will meet to discuss Exercise 3 & 4. Recording of this session

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.

Speaking of Davenport's discussion on p. 54, you may find this letter from Niels Henrik Abel commenting on "Legendre's 1.08366" interesting.

5–6 exercises will be given. 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) Proposed solutions
• Exercise 2 Proposed solutions
• 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. Proposed solutions
• Exercise 4 Proposed solutions

The syllabus for the course is as defined by the lectures. 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.

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 (chosen)
2. Ramanujan primes
3. Skewes's number and sign changes in \( \pi(x)-\operatorname{li}(x) \) (chosen)
4. General distribution of nontrivial zeros of \(\zeta(s)\) (chosen)
5. Zeros on the critical line, including density results (chosen)
6. Prime gaps (should include something about the relevance of the zeros of zeta) (chosen)
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 (chosen)
10. Dirichlet's divisor problem, including Voronoi's summation formula
11. Elementary sieve methods and Brun's theorem on twin primes (chosen)
12. Voronin's universality theorem and value distribution of the Riemann zeta function (chosen)
13. Lagarias's version of Guy Robin's criterion
14. The Beurling–Nyman condition for RH
15. Li's criterion for RH (chosen)
16. The Bohr–Cahen formulas for abscissas of convergence and the growth of \(\sum_{n\le x} \mu(n)\). (chosen)
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) (chosen)
18. Approximations of \(\zeta(s)\), including the approximate functional equation
19. The Riemann–Weil (or Guinand–Weil) explicit formula
20. Elementary proof of the prime number theorem (chosen)
21. Siegel zeros (chosen).

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.

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. (NB! Comment added on January 26, in response to input from the reference group: If you find that your favorite topic has been chosen by someone else, please contact me so that we can find a suitable adjustment, say that we find another topic that is very close, or that you may present different aspects of the same topic.)

• Johanne Krogholm Sand <johaksa(at)stud(dot)ntnu(dot)no>
• Jasper Steinberg <jasper.steinberg(at)ntnu(dot).no>
• Denis Zelent <denisze(at)stud(dot)ntnu(dot)no>

The first meeting of the reference group took place on January 26 (see here for the minutes from that meeting).

• Oral presentations will take place on May 8 in Kjel 22.
• Oral examination will take place on May 9 in Elektro E/F F304.

Here is the schedule for the oral exam.