Topics in Analytic Number Theory: Riemann's zeta function and beyond

Exam: 18th December 9-14 in Simastuen Exam schedule

In the first part of this course we will cover some aspects of the analytic the theory of the Riemann zeta function such as distribution of zeros, moments and the growth of zeta on the critical line. The second part of the course will cover some aspects of the distribution of multiplicative functions along with Brun's theorem on twin primes.

Prerequisites: MA3150 Analytic Number Theory

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

Markus Valås Hagen
Office: Room 954, Sentralbygg 2
Email: markus [dot] v [dot] hagen [at] ntnu [dot] no

• Lectures: Mondays 14:15-16:00 in 656 Simastuen.
• Problem Session: Fridays 13:15-14:00 in 656 Simastuen.

We will mostly follow:

• Lectures on the Riemann Zeta function by H. Iwaniec.
• The notes Topics in Analytic Number Theory by Winston Heap.
• The Theory of the Riemann Zeta-function 2nd edition by E.C. Titchmarsh.
• The Distribution of Prime Numbers by Dimitris Koukoulopoulos.

Lecture 1: Lecture Notes Lecture 1 (Iwaniec Chapter 2 & Chapter 5) Some motivation and overview on what we will try to cover. The Euler-Maclaurin formula. Estimating exponential sums. Approximation formula for Riemann zeta inside the critical strip by a truncated Dirichlet series.

Lecture 2: Lecture Notes Lecture 2 (Iwaniec 13.1 and 13.2, Heap 4.1 and 4.2) Dirichlet polynomials. Montgomery-Vaughan mean value theorem for Dirichlet polynomials. Second moment of zeta on the critical line. Discussion about higher moments. Equivalence between growth of moments and Lindelöf Hypothesis.

Lecture 3: Lecture Notes Lecture 3 (Titchmarsh 9.9 and 9.15) The zeroes of \( \zeta(s) \) in the critical strip: some repetition about \( N(T) \). Definition of \( N(\sigma, T) \) (the number of zeroes \( \rho \) with \( \text{Re}(\rho)>\sigma \)). Proof of \( N(\sigma,T) \ll T, \frac{1}{2} < \sigma < 1\) by using Littlewood's lemma. Mollification of zeta. Start of proof for \( N(\sigma,T)\ll T^{4\sigma(1-\sigma)+\varepsilon} \).

Lecture 4: Lecture Notes Lecture 4 (Titchmarsh 9.15 and 10.5) Continuation of \( N(\sigma,T)\ll T^{4\sigma(1-\sigma)+\varepsilon} \). The zeroes of \( \zeta(s)\) on the critical line, \( N_0(T) \). Start of the proof that there is an infinite amount of zeroes on the critical line: proof by counting sign changes of Hardy's Z-function.

Lecture 5: Lecture Notes Lecture 5 (Titchmarsh 10.5 and 9.9) We finish the proof of \(N_0(T) \to \infty \) as \( T \to \infty \). Then we prove the convexity bound for \( \zeta \) by studying the function \( \mu(\sigma)= \inf_{\xi} \{\xi: |\zeta(\sigma+it)| = O(|t|^{\xi}) \} \) and in particular its convexity. Solution to LH implies \(N(\sigma,T+1)-N(\sigma,T) = o(\log T)\) for \(\sigma>\frac{1}{2}\).

Lecture 6: Lecture Notes Lecture 6 Extreme values of \(|\zeta(\tfrac12+it)|\) by means of Soundararajan's resonance method.

Lecture 7: Counting \(y\)-rough and \(y\)-smooth numbers (Ch. 14 in Koukoulopoulos). We start by reviewing the sieve of Erathostenes–Legendre on pp. 28–29 of Koukouloplous, which is illuminating both for our study of rough numbers and for our approach to Brun's theorem (Lecture 9). We then prove the basic formula (14.2) in Koukoloupolos. We first use this formula to establish a simplified version of Thm. 14.2. We then proceed to the problem of estimating the number of \(y\)-rough numbers not exceeding \(x\), again starting from formula (14.2). We will see the appearance of the Buchstab function, and similarly we will see the appearance of the Dickman function in the corresponding problem for \(y\)-smooth numbers. Mertens's third bound (see Thm. 3.4 (c) along with the Erathostenes–Legendre formula suggests what the asymptotics of the Buchstab function should be. This asymptotics and the curious oscillatory behavior of the Buchstab function will be studied in detail in the exercises.

Lecture 8: The Erdős–Kac theorem (Ch. 15 in Koukoulopoulos). The Erdős–Kac theorem, published in 1940, is fundamental and historically important, as it marks the beginning of probabilistic number theory. We let \(\omega(n)\) denote the number of distinct prime divisors of a positive integer \(n\). Given a large positive number \(x\), we consider the probability that \(\omega(n)=k\) to be \(1/[x]\) times the number of \(m\) in \([1,x]\) such that \(\omega(m)=k\). The Erdős–Kac theorem asserts that \((\omega(n)-\log\log x)/\sqrt{\log\log x}\), in distribution, tends to a Gaussian random variable of mean \(0\) and variance \(1\). The intuition behind this result is that the Bernoulli random variables \(B_p\), where \(B_p(n)=1\) if \(p|n\) and \(0\) otherwise, are approximately independent in a sense that can be made quantitatively precise. As the random variable in our case is a sum of the \(B_p\), we might expect that the Erdős–Kac theorem may follow from an appropriate version of the central limit theorem, along with the Mertens estimate \(\sum_{p\le x} 1/p=\log\log x +O(1)\). The original proof of Erdős–Kac used the work of Viggo Brun (to be discussed in the next lecture). We will follow the proof of Billingsley using the method of moments, as discussed in Koukoulopoulos. Two other proofs can be found in Tenenbaum's book.

Lecture 9: Brun's theorem on twin primes (Ch. 17 in Koukoulopulos) We deduce Viggo Brun's bound \(\pi_2(x)\ll x (\log\log x)^2/(\log x)^2\), where \(\pi_2(x) \) is the number of positive integers \(n \le x\) such that both \(n\) and \(n+2\) are primes. We begin by looking at Legendre's work on the sieve of Erathostenes. We then proceed to explain how Viggo Brun was able to revolutionize Legendre's argument by making a finer study of the underlying inclusion-exclusion principle. We obtain thus a bound for \(\pi_2(x) \) with drastically fewer terms. The associated remainder term then consists of two parts, and the crux of the matter is to find the right balance between these two parts. The Cramér model suggests that Brun's estimate is off by a factor \( (\log\log x)^2\) from the truth; a refined version of this argument due to Granville leads to a precise conjecture about the asymtpotics of \(\pi_2(x)\). Here are notes from the lecture

Exercises file: exercises.pdf.

• Homework from Lecture 1: Choose some of the problems from 1-9 from the pdf.
• Homework from Lecture 2:
  • Look at the proof of the corollary regarding Dirichlet polynomial approximation of \( \zeta(s) \) inside the critical strip (proof in lecture notes).
  • Finish the proof of the Montgomery–Vaughan mean value theorem, i.e. prove the opposite inequality to the in class: \[ \int_T^{2T} \left| \sum_{n \leq N} \frac{a_n}{n^{it}} \right|^2 \, \mathrm{d}t \geq (T-N)\sum_{n \leq N} |a_n|^2 + O\left(N \sum_{n \leq N} |a_n|^2 \right) \] Hint: WLOG assume \( N \ll T\), say \( N < T/10 \). Then mimic the proof that we did in class for the \( \leq \)-inequality, but choose another \(f(t)\) so that \( \int_T^{2T} \left|\sum_{n \leq N} \frac{a_n}{n^{it}}\right|^2 \, \mathrm{d}t \geq \int_{\mathbb{R}} \left|\sum_{n \leq N} \frac{a_n}{n^{it}}\right|^2f(t) \, \mathrm{d}t \) and \(F(1) = T-N \) instead (where \(F(x) = \int_{\mathbb{R}} x^{-it}f(t)\,\mathrm{d}t \)).
  • Look at (and understand) the proof that \( I_k(T) \ll T^{1+\varepsilon} \forall \varepsilon>0, k \in \mathbb{N} \iff \mathrm{Lindelöf\, Hypothesis} \) at the end of the lecture notes.
  • Play around with the proof that the \( I_k(T) \ll T^{1+\varepsilon} \forall \varepsilon>0, k \in \mathbb{N} \implies \mathrm{Lindelöf\, Hypothesis} \). How high moments do we have to know sharp upper bounds for to beat the best bound we have towards LH, namely that \( \zeta(\tfrac12+it) \ll_{\varepsilon} t^{13/84+\varepsilon}\)?
  • If you have some extra time, take a look at the "Week 4"-exercises in the PDF.
• Homework from Lecture 3:
  • Let \(t\) be so that \(\zeta(a+it)\neq 0 \) for all \(0\leq a \leq 1\). For \(\tfrac12\leq \sigma\leq 1\), define \( S(\sigma,t) = \frac{1}{\pi} \mathrm{arg}\zeta(\sigma+it) \), where the argument is obtained by continuous variation along the line segments joining the points \(2\), \(2+it\) and \(\sigma+it\). Prove that \(S(\sigma,t) = O(\log t)\) (Hint: Take a look at how you proved \(S(t)=O(\log t)\) in the earlier analytic number theory course.)
  • Carry out the missing details in the proof of Littlewood's theorem on \(\int_{\sigma_0}^1 N(\sigma,T)\,\mathrm{d}\sigma\).
  • Prove that \( \int_{T}^{2T} |\zeta(\sigma+it)|^2\,\mathrm{d}t \sim T\zeta(2\sigma) \) for \(\sigma>1\).
  • (Pretty open exercise) Assume the Riemann Hypothesis is true. Derive a conjecture for the range of \(y\) for which a prime number theorem in short intervals, i.e. \( \psi(x+y)-\psi(x) \sim y \), should hold. Can you prove your conjecture (assuming Riemann Hypothesis)? What if you weaken your assumption to a partial RH (like no non-trivial zeroes \( \rho \) with \(\mathrm{Re}(\rho)>3/4\))? What if you weaken your assumption even more to only a zero density hypothesis? (Hint: Start with the explicit formula \(\psi(z)=z+\sum_{|\rho|\leq T}^* \frac{z^\rho}{\rho}+ \dots\).)
• Homework from Lecture 4:
  • Do Exercise 18 and 19 from the exercises.pdf-file.
  • Prove that uniformly for \(\frac{1}{2}\leq \sigma \leq \frac{3}{4} \), there is a constant \(A>0\) such that \[\int_{T}^{2T} |\zeta(\sigma+it)|^2\, \mathrm{d}t < AT\min \left(\log T, \frac{1}{\sigma-\frac{1}{2}} \right).\]
  • Recall Stirling's formula, which says that in any fixed strip \(\alpha\leq \sigma \leq \beta\), we have as \(t \to \infty\) that \[\log \Gamma(\sigma+it)=(\sigma+it-\tfrac12)\log(it)-it+\tfrac{1}{2}\log(2\pi) + O(t^{-1}).\] Write the classical functional equation of the Riemann zeta function in the form \[\zeta(s)=\chi(s)\zeta(1-s).\] Prove that \[\chi(s)=\left(\frac{2\pi}{t}\right)^{\sigma+it-\tfrac12}e^{i(t+\tfrac{\pi}{4})}\left(1+O\left(\frac{1}{t}\right)\right).\]
  • In the notation from the previous exercise, define \( \vartheta = -\frac{1}{2}\mathrm{arg}\chi(\frac12 +it) \) so that \(\chi(\frac12+it)=e^{-2i\vartheta(t)} \). Recall the definition of Hardy's \(Z\)-function: \( Z(t)=e^{i\vartheta(t)}\zeta(\tfrac12+it) = \chi(\frac12+it)^{-\frac12}\zeta(\frac12+it)\). Prove that \(t \in \mathbb{R} \implies Z(t) \in \mathbb{R}\) and that \( |Z(t)|=|\zeta(\frac12+it)| \).
• Homework from Lecture 5:
  • Let \( F\) be a real twice differentiable function such that \( F''(x)\geq r > 0 \) or \( F''(x)\leq -r < 0 \) for all \( x \in [a,b] \). Let \( G(x) \) be a real function such that \( \frac{G(x)}{F'(x)} \) is monotonic and \( |G(x)|\leq M \). Prove that \[ \left|\int_a^b G(x)e^{iF(x)} \, \mathrm{d}x \right| \leq \frac{8M}{\sqrt{r}}. \]
  • Let \( \mu(\sigma) = \inf \{ \alpha \in \mathbb{R} : |\zeta(\sigma+it)|=O\left(|t|^{\alpha+\varepsilon}\right) \} \). Here \(\varepsilon>0\), and should be thought of as arbitrary small. Prove that \( \mu(\sigma) \geq 0 \) for all \(\sigma\).
  • Assume that the Lindelöf Hypothesis holds true, i.e. that \(\mu(\tfrac{1}{2})=0\). Prove that for every \(\sigma>\tfrac{1}{2}\), we have \(N(\sigma,T+1)-N(\sigma,T)=o(\log T)\). (Hint: Apply Jensen's formula on \(\zeta(s)\) to a circle with center in \(2+it\) and radius \(\frac{3}{2}-\frac{\delta}{4}\). Apply Lindelöf's hypothesis to one of the sides of the equality.)
• Homework from Lecture 6:
  • Let \(R(t)=\sum_{n \leq N} \frac{r(n)}{n^{it}} \) with \( N \leq T^{1-\varepsilon}\). Assume that \(\Phi\) is a \(C^\infty\) function with compact support in \([1,2]\) such that \(0 \leq \Phi(t)\leq 1 \) always, and \(\Phi(t)=1\) for \(t \in (\frac{5}{4},\frac{7}{4})\). Denote the Fourier transform by \(\widehat{\Phi}(y)=\int_{\mathbb{R}} \Phi(t)e^{-ity}\,\mathrm{d}t\). Use \(\zeta(\tfrac12+it)=\sum_{k\leq T} \frac{1}{k^{1/2+it}} + O(T^{-1/2})\) to prove that \[\int_{\mathbb{R}} \zeta(\tfrac12+it)|R(t)|^2\Phi\left(\frac{t}{T}\right)\,\mathrm{d}t = T\widehat{\Phi}(0)\sum_{mk=n\leq N} \frac{r(m)\overline{r(n)}}{\sqrt{k}} + O\left(T^{1/2}\sum_{n \leq N} |r(n)|^2\right).\]
  • Let \(a,b:\mathbb{N} \to \mathbb{C}\) be two multiplicative functions. Show that the Dirichlet convolution \( (a*b)(n)=\sum_{d\mid n} a(d)b(n/d)\) is also a multiplicative function.
  • Given that \(\int_{T}^{2T} |\zeta(\tfrac12+it)|^{2k} \sim c_{k} T(\log T)^{k^2} \) for some constant \(c_k>0\) for \(k=1,2\), show that \[\mathrm{meas}\left\{t\in [T,2T] : |\zeta(\tfrac12+it)|\geq \sqrt{\frac{1}{2}\log T}\right\}\gg \frac{T}{(\log T)^2}.\] (Hint: Cauchy–Schwarz) (Bonus: What more can you say if you know the order of even higher moments than just the fourth?)
  • Let \(f(n)\geq 0 \) be an arithmetic function whose Dirichlet series \(F(s)=\sum_{n\geq 1} \frac{f(n)}{n^s} \) converges for \(\sigma>1\) and assume that \(F(\sigma)=\frac{1}{\sigma-1}+O(1)\) when \(\sigma \to 1^+\). Use Rankin's trick \(\mathbf{1}_{n \leq x} \leq (x/n)^{\alpha}\) to show that \[\sum_{n\leq x} \frac{f(n)}{n} \ll \log x. \] With the same notation, also show that the Rankin's trick only gives \[\sum_{n \leq x} f(n) \ll x\log x\] for the unweighted sum. Why is the latter inequality bad? When is Rankin's trick not sharp?
  • Let \(\tau(n)\) denote the number of divisors of \(n\). In this exercise we shall see an example on how we can transport a bound on \(\sum_{n \leq x} \frac{f(n)}{n}\) to a bound on \(\sum_{n \leq x} f(n)\). This is all based on the equality \(1 = \frac{\log(x/n)+\log n}{\log x}\). Use \(\log(x/n) \ll \frac{x}{n}\) and \(\log n = \sum_{d\mid n} \Lambda(d)\) to show that \[\sum_{n \leq x} \tau(n) = x\sum_{n \leq x} \frac{\tau(n)}{n} + \frac{1}{\log x}\sum_{n\leq x}\sum_{d \mid n} \tau(n)\Lambda(d) = x\sum_{n \leq x} \frac{\tau(n)}{n} + \frac{1}{\log x}\sum_{n \leq x}\sum_{d \leq x/n} \Lambda(d)\tau(nd).\] Then use a mixture of the subadditivity of \(\tau\), i.e. \(\tau(nd)\leq \tau(n)\tau(d)\), and the prime number theorem to prove that \[\frac{1}{\log x}\sum_{n \leq x}\sum_{d \leq x/n} \Lambda(d)\tau(nd)\ll \frac{x}{\log x} \sum_{n \leq x}\frac{\tau(n)}{n}.\] Conclude that \[\sum_{n\leq x}\tau(n) = x\sum_{n\leq x} \frac{\tau(n)}{n} + O\left(\frac{x}{\log x} \sum_{n \leq x} \frac{\tau(n)}{n}\right).\]
• Homework from Lecture 7: Exercises 14.10 (a)-(b) (hint: remember the functional equation of \(\Gamma\)) and 14.11 in Koukoulopoulos (in 14.11 (d) you should look at Exercise 5.4 (d) instead of (c)). For a more precise version of 14.11 (f), see Lemma 4 of Helmut Maier's paper. Here are sketches of solutions.
• Homework from Lecture 8: Exercise 15.2 in Koukoulopoulos. Here are sketches of solutions.
• Homework from Lecture 9: Exercise 17.3 (c) in Koukoulopoulos. Here is a sketch of the solution.

The syllabus for the course is as defined by the lectures. We 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 exam will be in two parts. For the last month of the course you will get assigned a topic to immerse yourself into. In addition to this we will have a classical oral exam part for ish 20 minutes. Both parts counts towards the grade. The list of topics is as follows (still to be updated):

1. Heath-Brown's twelfth power moment estimate for the Riemann zeta function Heath-Brown
2. Soundararajan's almost sharp upper bounds for moments of the Riemann zeta function, assuming RH Soundararajan. See also Harper's sharp upper bounds which removes the \( \varepsilon \) in Soundararajan's result.
3. [Connor] Soundararajan's and Radziwill's proof of Selberg's central limit theorem Soundararajan--Radziwill. See also Tao's blog post on the matter.
4. Montgomery's pair correlation conjecture Montgomery
5. Ingham's asymptotic for the fourth power moment of the Riemann zeta function (See chapter 4 of the Lecture Notes of Winston Heap above)
6. [Magnus] The Moments Conjecture Gonek--Hughes--Keating Keating--Snaith (see also Selberg's elegant paper on what is known today as the Selberg integral, which plays a key role in the random matrix computations Bemerkninger om et multipelt integral)
7. [Truls] Hoheisel's theorem: zero-free region + zero density estimates implies prime number theorem in short intervals (and thus bounds on the difference between consecutive primes) Chandrasekharan's book. See also Chapter 10.5 in Iwaniec--Kowalski for some historical background and Guth--Maynard for a very recent breakthrough.
8. van der Corput's method, exponential pairs and its applications to bounding the zeta function on the critical line. Lecture notes by Terence Tao
9. The resonance method of Soundararajan, and the long resonance method of Aistleitner–Bondarenko–Seip. See chapter 5 of Winston Heap's lecture notes, and also the original papers: Soundararajan, Aistleitner and Bondarenko--Seip.
10. [Thomas] The Bombieri–Vinogradov theorem from the large sieve. If Bombieri–Vinogradov feels like too much, one can also do the easier case: the Barban–Davenport–Halberstam theorem. Notes on the large sieve (Ignore section 8.2. The Barban–Davenport–Halberstam theorem is section 8.4). Bombieri--Vinogradov
11. [Lukas] Linnik's theorem on the least prime in arithmetic progressions. Outline of proof strategy, and some details.