Analytic number theory is a vibrant branch of mathematics concerned with applying techniques from analysis to solve number theory problems. You’ll learn about a rich collection of analytic tools that can prove results, such as the prime number theorem. The module also introduces the Riemann hypothesis, one of mathematics’s most famous unsolved problems. Before embarking on this module, you should complete a complex analysis module, like {Analytic number theory I (M823]}, covering topics such as the calculus of residues and contour integration.
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05 Oct 2024 |
Jun 2025 |
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| Registration closes 05/09/24 (places subject to availability) Click to register
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This module is expected to start for the last time in October 2030.
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?
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 one of the following modules:
Or one of the discontinued modules M826, M828 and M832.
We recommend Analytic number theory I (M823).
We also recommend that you’ve passed a module in complex analysis, such as our Complex analysis (M337).
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.
Qualifications
M829 is an optional module in our:
If you have a disability
The material contains small print and diagrams, which may cause problems if you find reading text difficult and you may also want to use a scientific calculator.
To find out more about what kind of support and adjustments might be available, contact us or visit our disability support pages.
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 and tutor group forums.
You’ll also be provided with printed course notes, which includes a narrative to accompany the module text, additional exercises and solutions.
Computing requirements
You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.
Any additional software will be provided or is generally freely available.
To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).
Our module websites comply with web standards, and any modern browser is suitable for most activities.
Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.
It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.
Materials to buy
Set books
- Apostol, T.M. Introduction to Analytic Number Theory Springer £46.99 - ISBN 9780387901633 This book is Print on Demand and can be ordered through any bookseller. Please allow at least 2 weeks for receipt following order.
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.
Students also studied
Students who studied this module also studied at some time:
Future availability
Analytic number theory II (M829) starts every other year – in October.
This page describes the module that will start in October 2024.
We expect it to start for the last time in October 2030.
How to register
To register a place on this module return to the top of the page and use the Click to register button.
Regulations
As a student of The Open University, you should be aware of the content of the academic regulations which are available on our
Student Policies and Regulations website.