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:
In this MSc module (and in Analytic number theory II (M829)), you will study number theory using techniques from analysis, particularly the convergence of series and the calculus of residues. Among the results proved in this module are:
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.
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.
You’ll get help and support from an assigned tutor throughout your module.
They’ll help by:
Online tutorials run throughout the module. While they’re not compulsory, we strongly encourage you to participate. Where possible, we’ll make recordings available.
Course work includes:
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 contained in the set book.
You’ll also have access to a module website, which includes:
Set books:
You can study this module on its own or use the credits you gain towards an Open University qualification.
M823 is an option module in our:
Analytic number theory I (M823) starts once a year – in October.
It will next start in October 2026.
We expect it to start for the last time in October 2031.
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.
If you’re studying this module on its own or as part of a postgraduate qualification, you should have:
We consider all applications but may ask you to complete an entry test.
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.
The material contains small print and diagrams, which may cause problems if you find reading text difficult.
To find out more about what kind of support and adjustments might be available, contact us or visit our disability support pages.
| Start | End | Register by | England fee |
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| 03 Oct 2026 | 30 Jun 2027 | 10 Sep 2026 | Not yet available* |
| *This start date is open for pre-booking, which means you can reserve your place ahead of the fees being confirmed. We’ll publish updated 2026/27 fees and funding information in late March 2026. |
There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.
If your income is not more than £25,000 or you receive a qualifying benefit, you might be eligible for help with some of these costs after your module has started.
There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.
This module may have an optional study weekend. There will be a charge to attend in person or online. If attending in person, you’ll also need to pay for your travel, and accommodation if you need it. More details will be available after the module has started.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to support your study costs. To find out more, see Postgraduate loans in England.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to help with your tuition fees. To find out more, see Postgraduate tuition fee loans in Northern Ireland.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to help with your tuition fees. To find out more, see Postgraduate loans in Scotland.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to support your study costs. To find out more, see Postgraduate loans in Wales.
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As additional affordability checks are required when processing joint loan applications, unfortunately, an instant decision cannot be given. On average, the processing time for a joint loan application is five working days from receipt of the required documentation.
Read more about Open University Student Budget Accounts (OUSBA).
Studying with The Open University can boost your employability. OU courses are recognised and respected by employers for their excellence and the commitment they take to complete. They also value the skills that students learn and can apply in the workplace.
Over 30,000 employers have used the OU to develop staff so far. If the module you’ve chosen is geared towards your job or developing your career, you could approach your employer to see if they will sponsor you by paying some or all of the fees.
You can pay part or all of your tuition fees upfront with a debit or credit card when you register for each module.
We accept American Express, Mastercard, Visa and Visa Electron.
We know that sometimes you may want to combine payment options. For example, you may wish to pay part of your tuition fee with a debit card and pay the remainder in instalments through an Open University Student Budget Account (OUSBA).
We know that sometimes you may want to combine payment options. For example, you may get support from your employer to pay part of your tuition fee and pay the remainder by credit or debit card.
For more information about combining payment options, contact an adviser.
Please note: your permanent address/domicile will affect your fee status and, therefore, the fees you are charged and any financial support available to you. The fee information provided here is valid for modules starting before 31 July 2026. Fees typically increase annually. For further information about the University's fee policy, visit our Fee Rules.
