Introductory description

Analytic number theory uses methods of real and complex analysis to study the distribution of interesting number theoretic objects. It is a huge subject, so to focus the course we will concentrate on questions about the distribution of primes, and mostly on those questions that can be attacked using the Riemann zeta function and related objects. This is the subject for which the zeta function and related functions were introduced and studied in the first place (by Euler, Dirichlet, Riemann, ...).

By ``the distribution of primes'', I mean questions like:

• how many primes are less than x ? (for an arbitrary large number x)

• how many primes are less than x and congruent to a modulo q, for given natural numbers a and q (possibly allowed to vary with x)?

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The course will cover some of the following topics, depending on time and audience preferences:
(1) Warm-up: The counting functions pi(x), Psi(x) of primes up to x. Chebychev's upper and lower bounds for Psi(x).
(2) Basic theory of the Riemann zeta function: Definition of the zeta function zeta(s) when Re(s) > 1, and then when Re(s) > 0 and for all s. The connection with primes via the Euler product. Proof that zeta(s) neq 0 when Re(s) geq 1, and deduction of the Prime Number Theorem (asymptotic for Psi(x)).
(3) More on zeros of zeta: Non-existence of zeta zeros follows from estimates for sum_{N < n < 2N} n^{it}. The connection with exponential sums, and outline of the methods of Van der Corput and Vinogradov. Wider zero-free regions for zeta(s), and application to improving the Prime Number Theorem. Statement of the Riemann Hypothesis.
(4) Primes in arithmetic progressions: Dirichlet characters chi and Dirichlet L-functions

Learning outcomes

By the end of the module, students should be able to:

• Have a good understanding of the Riemann zeta function and the theory surrounding it up to the Prime Number Theorem.
• Understand and appreciate the connection of the zeros of the zeta function with exponential sums and the statement of the Riemann Hypothesis.
• Demonstrate the necessary grasp and understanding of the material to potentially pursue further postgraduate study in the area.
• Consolidate existing knowledge from real and complex analysis and be able to place in the context of Analytic Number Theory

Indicative reading list

H. Davenport. Multiplicative Number Theory. Third edition, published by Springer Graduate Texts in Mathematics. 2000
A. Ivi'c. The Riemann Zeta-Function. Theory and Applications. Dover edition, published by Dover Publications, Inc.. 2003
H. Montgomery and R. Vaughan. Multiplicative Number Theory I. Classical Theory. Published by Cambridge studies in advanced mathematics. 2007
E. C. Titchmarsh. The Theory of the Riemann Zeta-function. Second edition, revised by D. R.
Heath-Brown, published by Oxford University Press. 1986

Subject specific skills

By the end of the module the student should be able to:

• consolidate existing knowledge from real and complex analysis in the context of Analytic Number Theory.
• use real variable methods such as those of Chebychev and Mertens, as well as summation by parts, to estimate certain sums over prime numbers.
• understand the Riemann zeta function and the theory surrounding it, including using the zeta function to prove the Prime Number Theorem.
• understand and appreciate the connection of the zeros of the zeta function with exponential sums and the statement of the Riemann Hypothesis.
• potentially pursue further postgraduate study in the area.

Transferable skills

The module will help to develop skills in understanding, assessing and constructing sophisticated logical arguments (especially of a quantitative nature), and presenting these clearly in writing. In the optional support classes, students will also have opportunities to work collectively and to present their arguments orally.