Number theory has roots in ancient history, but since the seventeenth century, it’s developed intensively using ideas from many branches of mathematics. Despite the subject’s maturity, there are still unsolved problems that are easy to state and understand – for example, is every even number greater than two the sum of two primes? In this module, and in Analytic number theory II (M829), you’ll study number theory using techniques from analysis, particularly the convergence of series and the calculus of residues. The module is based on readings from T.M. Apostol’s Introduction to Analytic Number Theory.
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:
- Is every even number greater than two the sum of two primes?
- Are there infinitely many ‘twin primes’ (primes differing by 2), such as (3, 5) or (101, 103)?
- Are there infinitely many primes of the form n 2 + 1?
- Does there always exist a prime between n 2 and (n + 1)2 for every integer n > 1?
In this MSc module (and in Analytic number theory II (M829)), you will study number theory using techniques from analysis, in particular, the convergence of series and the calculus of residues. Among the results proved in this module are:
- Dirichlet’s theorem on primes in an arithmetic progression, which states that there are infinitely many primes in a progression such as 1, 5, 9, 13, 17 …
- the law of quadratic reciprocity, which compares the solubility of the congruences x2 ≡ p(mod q) and x2 ≡ q(mod p), where p and q are primes.
This module is based on selected readings from the set book Introduction to Analytic Number Theory by T. M. Apostol. It covers most of the material in the first seven chapters, and part of Chapter 9.
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.
This module and Calculus of variations and advanced calculus (M820) are entry-level modules for the MSc in Mathematics (F04), and normally you should have studied one of them before progressing to the intermediate and advanced intermediate modules.
Note you must have completed this module before studying Analytic number theory II (M829).
Entry
If you’re studying this module on its own or as part of a postgraduate qualification, you should have:
- an honours degree in mathematics (minimum 2:2) or
- an honours degree in a subject with a high mathematical content.
We consider all applications but may ask you to complete an entry test.
Preparatory work
You should have a solid background in pure mathematics, with some experience in number theory and analysis. Pure mathematics (M208) (or equivalent) and Further pure mathematics (M303) (or other experience of analysis at third-year honours level) should provide adequate preparation.
Whatever your background, you should assess your suitability with our diagnostic quiz.
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 be provided with printed course notes covering the content of the module, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques that are contained in the set book.
You’ll also have access to a module website, which includes:
- a week-by-week study planner
- course-specific module materials
- audio and video content
- assessment details and submission section
- online tutorial access
- access to student forums.
You will need to obtain your own copy of the set book. Only the set book as printed by the publisher will be permitted in the examination, and not a version you have printed yourself.
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.
Future availability
Analytic number theory I (M823) starts once a year – in October.
This page describes the module that will start in October 2025.
We expect it to start for the last time in October 2027.