PMH9 Analytic Number Theory

General Information

This page relates to the Pure Mathematics Honours course "Analytic Number Theory".

Lecturer for this course: Dzmitry Badziahin.

For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.

Lecturer Contact Information

The easiest way to contact me is via email. Also, quite often you can find me in the office C634.

Class Times

The lecture times for the course are

Monday 11am in C830;

Thursday 10am in C830.

Exercise Sheets

Lecture Notes

Here are the part 1, part 2 and part 3 of the lecture notes for this course. They will gradually add up throughout the course.

Initially I wrote them for myself, so in some places they may not be easy to read. I hope that they will be useful.

Course Outline

The aim of this course is to show how analytical methods can be used to tackle problems in number theory. Famous examples include Prime Number Theorem about the asymptotic density of primes and Dirichlet Theorem about prime numbers in arithmetic progressions. We will see why zeroes of the Riemann zeta function are so important that one million dollars problem is related with them.

Another area which will be covered is Diophantine approximation. It investigates various approximational properties of real numbers, points in Rn or of elements in other metric spaces. We will see that different real numbers can be approximated by rationals with different efficiency, find out how to compute best rational approximations to a given real number and what is the approximational behaviour of a generic real number.

The approximate list of covered topics is:

Distribution of primes Euler's proof of infinitude of prime numbers; Riemann zeta function and its analytic continuation; Prime Number Theorem about the number of primes up to n; Estimate of the error term in the prime number theorem; Riemann conjecture; Dirichlet characters and L-functions; Dirichlet's theorem about the infinitude of prime numbers in arithmetic progressions;

Diophantine approximation Approximation of irrational numbers by rationals, Dirichlet theorem; Continued fractions and best approximants; Khintchine's theorem about the approximational behaviour of generic real numbers. Approximational properties of quadratic irrationals; Liouville's theorem.



The list may appear to be quite ambitious. If we do not have time, a couple of items at the end of this list will be skipped.

Assessment

There will be two assignments, each worth 20%, and the final exam worth 60%.

Assignment 1 is now available online and is due before midnight on Thursday September 12 (Week 6) .

. Assignment 2 is now available online and is due before midnight on Thursday October 31 (Week 12).

In the event of special considerations, the maximum possible extension will be 7 days, to allow for assignments and feedback to be returned the following week.

The final exam, worth 60%, will be 2 hours plus 10 minutes reading time. No notes or calculators of any type are allowed at the exam.

Timetable