# The Riemann Zeta Function

We will cover some aspects of the analytic the theory of the Riemann zeta function such as distribution of zeros, moments, extreme values, conjectures arising from random matrix theory, the Selberg central limit theorem.

Prerequisites: MA3150 Analytic Number Theory

## Lecturers

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

## Schedule

- Lectures: Tuesday 10:15–12:00, Room 656 (Simastuen)
- Problem Session: Friday 10:15–12:00, Room 656 (Simastuen) on January 19, then Room 822 from January 26.

## Textbook

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.

## Lectures

**Lecture 1, 16.01:** (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, 23.01:** (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. Equivalence between growth of moments and Lindelöf Hypothesis.

**Lecture 3, 30.01:** (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, 06.02:** (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, \( N_0(T) \). There is an infinite amount of zeroes on the critical line, proof by counting sign changes of Hardy's Z-function. Remarks on \( N_0(T) \): the role of mollification, Selberg, Levinson, Conrey…

**Lecture 5, 13.02:** (Titchmarsh 10.5 and 9.9) We finish the proof of \(N_0(T) \to \infty \) as \( T \to \infty \). Convexity bound for \( \zeta \): the function \( \mu(\sigma)= \inf_{\xi} \{\xi: |\zeta(\sigma+it)| = O(|t|^{\xi}) \} \). We then start looking at the "fine scale" statistics of zeroes. \( S_1(T) = \int_0^T S(t) \, \text{d}t = O(\log T ) \) by Littlewood's lemma.

**Lecture 6, 20.02:** (Titchmarsh 9.9, Heap 6.5.1) We finish the proof of \( S_1(T) = \int_0^T S(t) \, \text{d}t = O(\log T ) \). \( S(T) \) changes sign infinitely many times. Some loose philosophical explanation on the central limit theorem of Selberg.

**Lecture 7, 27.02:** We start our study of the paper "Selberg's central limit theorem for \(\log|(1/2+it)|\)" by Maksym Radziwiłł and Kannan Soundararajan.

**Lecture 8, 05.03:** We will finish our discussion of the Selberg central limit theorem. We follow Terence Tao's blog post "254A, Notes 2: The central limit theorem" to get Selberg's theorem from what we have proved about the moments. See Thm. 13 and Thm. 25 in this blog post.

**Lecture 9, 12.03:** We will look at Hugh L. Montgomery's paper "The pair correlation of zeros of the zeta function" and discuss his pair correlation conjecture.

**Lecture 10, 19.03:** We discuss Montgomery's pair correlation conjecture based on the paper "Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series" by D. A. Goldston and S. M. Gonek. We focus on the probabilistic heuristics behind Polignac's conjecture (the strong twin prime conjecture).

## Exercises

## Syllabus and requirements for the examination

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.

## Exam

To be announced.