Analytic number theory is about applying techniques from analysis to tackle problems in number theory. In this module, the sequel to Analytic number theory I (M823), you’ll learn about Gauss sums, primitive roots, Dirichlet series, Euler products, Dirichlet L-functions, the gamma function and the Riemann zeta function. Highlights include a proof of the celebrated prime number theorem and an introduction to the Riemann hypothesis, one of the greatest unsolved problems in mathematics, finishing with applications of analytic number theory to integer partitions. Alongside acquiring advanced mathematical skills, you’ll enhance your ability to argue logically and communicate complex mathematical ideas.
What you will study
The Greeks were the first to classify the integers and it is to them that the first systematic study of the properties of the numbers is attributed. But after about AD 250 the subject stagnated until the seventeenth century. Since then there has been intensive development, using ideas from many branches of mathematics. There are a large number of unsolved problems in number theory that are easy to state and understand – for example:
This module (and the preceding module Analytic number theory I (M823)) are about the application of techniques from analysis in solving problems from number theory. In particular, you’ll learn about the prime number theorem, which estimates how many prime numbers there are less than any given positive integer. You’ll also find out about the Riemann hypothesis, one of the most famous unsolved problems in mathematics. To understand these topics, you’ll study certain rich classes of functions that are analytic in parts of the complex plane, among them the Riemann zeta function, which is the subject of the Riemann hypothesis.
This module is based on Chapters 8-14 of the set book Introduction to Analytic Number Theory by T. M. Apostol.
You will learn
Successful study of this module should enhance your skills in understanding complex mathematical texts, working with abstract concepts, thinking logically and constructing logical arguments, communicating mathematical ideas clearly and succinctly, and explaining mathematical ideas to others.
Entry
You must have passed:
Preparatory work
You should have studied an undergraduate course in complex analysis covering topics such as the calculus of residues and contour integration. Complex analysis (M337) should provide adequate preparation.
All teaching is in English and your proficiency in the English language should be adequate for the level of study you wish to take. We strongly recommend that students have achieved an IELTS (International English Language Testing System) score of at least 7. To assess your English language skills in relation to your proposed studies you can visit the IELTS website.
Study materials
What's included
You’ll have access to a module website, which includes:
- a week-by-week study planner
- course-specific module materials
- audio and video content
- a specimen exam paper with solutions
- assessment details and submission section
- online tutorial access
- access to student forums.
You’ll also be provided with printed course notes, which includes a narrative to accompany the module text, additional exercises and solutions.
Teaching and assessment
Support from your tutor
Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:
- Marking your assignments (TMAs) and providing detailed feedback for you to improve.
- Guiding you to additional learning resources.
- Providing individual guidance, whether that’s for general study skills or specific module content.
The module has a dedicated and moderated forum where you can join in online discussions with your fellow students. There are also online module-wide tutorials. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part. If you want to participate, you’ll likely need a headset with a microphone.
Assessment
The assessment details can be found in the facts box.