Analytic number theory studies the distribution of 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 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

Dimitris Koukoulopoulos, "The Distribution of Prime Numbers", Graduate Studies in Mathematics 203, American Mathematical Society, Providence RI, 2019.

We will mainly follow Koukoulopoulos's book, but you may also be interested in taking a look at the classic book "Multiplicative Number Theory" by Harald Davenport.

January 8 to January 30 (weeks 2–5):

• Wednesday 12:15–14:00, F4
• Thursday 08:15–10:00, R41

February 5 to 26 (NB! This link may be helpful for finding your way):

• Wednesday 12:15–14:00, MA23
• Thursday 08:15–10:00, MA23

NB! February 27 - April 10:

• Wednesday 12:15–14:00, F3
• Thursday 08:15–10:00, Simastuen 656 (Sentralbygg 2)

The first lecture will take place on January 8. The lectures will end a few weeks before Easter so that you have enough time to prepare your oral presentation (see below).

• Lecture 1, January 8: A short historical introduction. The infinitude of prime numbers; Euler's argument (see Lemma 2.3 in Koukoulopoulos); heuristics in favor of Gauss's conjecture (which later became the prime number theorem); the research program initiated by Dirichlet (1837) and Riemann (1859), which is what we will be mainly concerned with.
• Lecture 2, January 9: Koukoulopoulos, Chapter 1. Notation for asymptotic estimates, summation by parts, the Euler–Mascheroni constant \(\gamma\), relation between \(\pi(x)\) and the Chebyshev function \(\theta(x)\), Stirling's formula for \(n!\).
• Lecture 3, January 15: Meromorphic continuation of \( \zeta(s) \). (See this note for both Abel summation and the Euler formula applied to \( \zeta(s) \).) Euler's totient function \(\phi(n)\) (see Koukoulopoulos, Chapter 2), the Möbius function \(\mu(n)\), multiplicative functions.
• Lecture 4, January 16: Möbius inversion, Dirichlet convolution, the von Mangoldt function \(\Lambda(n)\) (see Koukoulopoulos, Chapter 3). Sums of arithmetic functions; Chebyshev's bounds on \( \pi(x) \) and Mertens's theorem on the asymptotics of \(\sum_{p\le x} p^{-1}\) (see Koukoulopoulos, Chapter 3 and this note for a short route to these results; see also here for an interesting account of Mertens's theorems).
• Lecture 5, January 22: The hyperbola method: Dirichlet's divisor problem (see Koukoulopoulos, Chapter 3), Dirichlet series and Dirichlet convolution, the convergence abscissas for Dirichlet series, the Riemann zeta function \(\zeta(s)\)and its Euler product, the logarithmic derivative of zeta, Möbius inversion and the reciprocal of zeta (see Chapter 4 of Koukoulopoulos).
• Lecture 6, January 23: Remarks on Exercise 1. Our plan for proving PNT via the explicit formula, Perron's formula as a first step to establish the explicit formula, the Fourier duality between primes and zeros of zeta (see Koukokoulopoulos, Chapter 5 and 7). We have now established the formula \(\psi(x)=-\frac{1}{2\pi i}\int_{1+\frac{1}{\log x}-iT}^{1+\frac{1}{\log x}+iT} \frac{\zeta'(s)}{\zeta(s)}\frac{x^s}{s} ds + O(\frac{x}{T}\log^2 x)+O(\log x)\). To get what we want from this formula, we need to know more about the properties of \(\zeta(s)\) as a meromorphic function. The next lectures will establish the key facts that we need.
• Lecture 7, January 29: 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, Euler's reflection formula, Legendre's duplication formula. See Winston Heap's note on the Gamma function and also page 73 of Davenport where the salient facts required about gamma are neatly summarized.
• Lecture 8, January 30: We finish our discussion of the gamma function by establishing the general version of Stirling's approximation, which among other things yields the exponential decay of gamma in the vertical direction. We next go back to zeta and start our dicussion of the functional equation \(\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)=\pi^{-(1-s)/2}\Gamma(\frac{1-s}{2})\zeta(1-s)\). We observe what it entails in terms of zeros of zeta (trivial and nontrivial zeros) and also use our new version of Stirling to show that \(\zeta(\sigma+it)\) grows as \(|t|^{-\sigma+\frac12}\) along vertical lines when \(-\sigma\ge \varepsilon > 0\). We prepare for the proof of the functional equation by noting that \(e^{-\pi x^2}\) is self dual and proving the Poisson summation formula \(\sum_{n\in \mathbb Z} f(n)=\sum_{n\in \mathbb Z} \widehat{f}(n)\).
• Lecture 9, February 5: We prove the functional equation (see p. 63 in Koukoulopoulos). We then move on to Lemma 8.2 in Koukoulopoulos which, along with what we have found about the gamma function, is what we need to arrive at the explicit formula for \(\Psi(x)\). You may note that we take a shorter route than Davenport who at this point has a discussion of entire functions of order 1, the product representation of \(\xi(s)\) (see Exercise 8.7 in Koukoulopoulos), and the Riemann-von Mangoldt formula. The latter result – obtained by a beautiful combination of the functional equation, the argument principle, and Stirling's approximation – is in our approach replaced by the crude bound \( N(T+1)-N(T)=O(\log T) \), which we deduce from Jensen's formula.
• Lecture 10, February 6: We finish the proof that \(\zeta'(s)/\zeta(s)=\sum_{|\gamma-t|\le 1} \frac{1}{s-\rho}+O(\log(|s|+2))\) when \(-1\le \sigma \le 2\) and, say, \(|t|\ge 1\). (Remark: You may want to check the proof of this result in Davenport pp. 98–99, which is technically simpler and conceptually appealing. However, we then need to establish the representation of \(\xi(s)\) as an Hadamard product.) We then proceed to deduce the explicit formula for \(\psi(x)\), as expressed in Thm. 5.1 in Koukoulopoulos, from the formula established in Lecture 6. (Remarks on Exercise 2 postponed.)
• Lecture 11, February 12: 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 \(\psi(x)=x+O(x^{\Theta})\). This shows that we could not have an error term of size essentially smaller than \(\sqrt{x}\) in the prime number theorem. We establish next the zero-free region of de la Vallée Poussin (see Thm. 8.3 in Koukoulopoulos); we will use the classical "3-4-1 inequality" (see Ch. 13 in Davenport). We then show how this leads to a proof of the prime number theorem with remainder term \(O(x e^{-c\sqrt{\log x}})\) for some constant \(c>0\). (See Ch. 8 in Koukoulopoulos.)
• Lecture 12, February 13: Remarks on Exercise 2. We establish the Riemann–von Mangoldt formula for \(N(T)\) which we get from the functional equation, the argument principle, and Stirling's formula (see Ch. 15 in Davenport).
• Lecture 13, February 19: We start preparing for Dirichlet's theorem on primes in arithmetic progressions by considering some facts about finite Abelian groups, including the character group and orthogonality of the characters. We define the \(\phi(k)\) Dirichlet characters modulo \(k\), which may be viewed as the multiplicative characters modulo \(k\) as opposed to the additive characters modulo \(k\) that you may know from Fourier analysis (see Ch. 9–10 in Koukoulopoulos).
• Lecture 14, February 20: Examples of characters: The characters modulo \(q\) for \(q=2,3,4,5,6,7,8\), the Legendre symbol. Primitive and imprimitive characters, Dirichlet \(L\)-functions. (see Ch. 10 in Koukoulopoulos).
• Lecture 15, February 26: We begin by explaining how we plan to prove PNT for arithmetic progressions; see formula (9.3) in Koukoulopoulos. Here we expect that \(\chi_0\) will yield the main contribution to the sum. To clarify the contribution from the sum over the remaining caharcters, we will act similarly as we did when dealing only with zeta. But we need more information about characters and in particular primitive characters. To this end, we look at the interaction between additive and multiplicative characters; Gauss sums (see in particular Thm. 10.4), Fourier representation of primitive characters (Thm. 10.3) with an associated extension of the Poisson summation formula (Thm. 10.5).
• Lecture 16, February 27: We now establish the functional equation that relates \(L(s,\chi)\) to \(L(1-s,\overline{\chi})\) for primitive characters \(\chi\) (see Thm. 11.1 in Koukoulopoulos). We then record some consequences of the functional equation for \(L\)-functions associated with primitive Dirichlet characters. We prepare for the proof of the explicit formula for the Chebyshev function \(\psi(x, \chi)\) by establishing appropriate estimates for \(L'(s,\chi)/L(s,\chi)\).
• Lecture 17, March 5: We prove the explicit formula for the Chebyshev function \(\psi(x, \chi)\). We now aim to establish a zero-free region for \(L(s,\chi)\) so that we can obtain our desired PNT for arithmetic progressions. We begin with the simpler result that \(L(1,\chi)\neq 0\) when \(\chi\) is a nonprincipal character modulo \(q\). This implies Dirichlet's classical theorem in the form \(\sum_{p\equiv a (\operatorname{mod} q)}1/p=\infty\) whenever \((a,q)=1\). (See the last part of Ch. 4 in Davenport for this argument.)
• Lecture 18, March 6: We establish a zero-free region for \(L(s,\chi)\). We begin by finding out that the "3-4-1 inequality" works as before for complex \(\chi\) and also for real \(\chi\) when the zeros are not too close to the real line. We then use a similar argument to clarify the possibility of zeros close to 1. We end up proving the following theorem: There is an absolute constant \(c\) such that the following holds. If \(\chi\) is a complex character, then there are no zeros in the region \( \sigma\ge 1-c/\log(q(|t|+1))\). If \(\chi\) is a real character, there can be only one zero in this region, and this exceptional zero must be real and simple. Taking Landau's theorem for granted (see next lecture), we establish Thm. 12.4 in Koukoulopoulos, which clarfies the effect of a possible exceptional zero close to \(1\).
• Lecture 19, March 12: We prove the important theorem of Landau (see Thm. 12.5 in Koukoulopoulos) which asserts that the exceptional zeros "repel" each other. We then proceed to prove a weak version of Theorem 12.8 in Koukoulopoulos and a corresponding weak version of the Siegel–Walfisz theorem.
• Lecture 20, March 13 (final ordinary lecture): We discuss the final profound point regarding zeros in arithmetic progressions, namely Siegel's theorem (Theorem 12.9 in Koukoulopoulos). We then see how this combined with what we have proved before yields the full Siegel–Walfisz theorem. (Here we follow pp. 128–131 in Davenport.)
• March 19: We discussed Exercise 4 and the related problem K8.6 b). Note that the conditional bound used for \(\log s\) in the proof K8.6 b)(see K8.5 b)), also yields the following basic fact: The Riemann hypothesis implies the Lindelöf hypothesis.
• March 20: We discuss selected problems exercises 3 and 5.

5–6 exercises will be given.

• Exercise 1: Koukoulopoulos 1.1, 1.2, 1.6, 1.7 (see also 2.7), 2.1, 2.2, 2.4 (a), 2.5. Sketch of solutions
• Exercise 2: Koukoulopoulos 3.1, 3.3, 3.4, 3.6, 3.14, 4.8, 4.9, 5.2. Sketch of solutions
• Exercise 3
• Exercise 4 (see Theorem 6.2 and Theorem 6.3 in Koukoulopoulos)
• Exercise 5: Koukoulopoulos: 8.6 b), 9.5, 9.6 a), 10.2 b) and c), 11.2, 12.2 a).

The syllabus for the course is as defined by the lectures, which are planned to cover at least the first 12 chapters of Koukoulopoulos's book. 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. Skewes's number and sign changes in \( \pi(x)-\operatorname{li}(x) \)
2. General distribution of nontrivial zeros of \(\zeta(s)\)
3. Zeros on the critical line (chosen)
4. Cramér's probabilistic model for the distribution of primes (chosen)
5. Prime gaps (chosen)
6. The Lindelöf hypothesis and the density hypothesis
7. Extreme values of the Riemann zeta function
8. Mean value theorems - results and conjectures
9. Dirichlet's divisor problem, including Voronoi's summation formula
10. Elementary sieve methods and Brun's theorem on twin primes (chosen)
11. Voronin's universality theorem and value distribution of the Riemann zeta function (chosen)
12. Lagarias's version of Guy Robin's criterion
13. The Beurling–Nyman condition for RH
14. Li's criterion for RH
15. The Riemann–Weil (or Guinand–Weil) explicit formula
16. Elementary proof of the prime number theorem (chosen)
17. Character sums (chosen)
18. Siegel zeros (chosen)
19. Montgomery's pair correlation conjecture (chosen)
20. Roth's theorem on arithmetic progressions (chosen).
21. The Green–Tao and Szemerédi theorems on arithmetic progressions (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! 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 may find another topic that is very close, or that you may present different aspects of the same topic.)

Please notify me by email or in class no later than March 14 which topic you have chosen for your presentation.

• Hans Magnus Bukholm
• Torgeir Keun Lysen
• Moritz Melcher

• Minutes from meeting January 28

• April 28, : Oral presentations of chosen topics
• April 29, Room 956 Sentralbygg 2: Oral examination

Here is the schedule for April 28–29. The presentations on Monday April 28 are open to everyone. A light lunch will be served at noon. You are all encouraged to attend the whole event!