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Analytic number theory I

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.

Qualifications

M823 is an option module in our:

Module

Module code
M823
Credits

Credits

  • Credits measure the student workload required for the successful completion of a module or qualification.
  • One credit represents about 10 hours of study over the duration of the course.
  • You are awarded credits after you have successfully completed a module.
  • For example, if you study a 60-credit module and successfully pass it, you will be awarded 60 credits.
30
Study level
Across the UK, there are two parallel frameworks for higher education qualifications, the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ) and the Scottish Credit and Qualifications Framework (SCQF). These define a hierarchy of levels and describe the achievement expected at each level. The information provided shows how OU postgraduate modules correspond to these frameworks.
OU Postgraduate
SCQF 11
FHEQ 7
Study method
Distance learning
Module cost
See Module registration
Entry requirements

Find out more about entry requirements.

Module details Entry requirements Module registration Study materials

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).

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.

Course work includes

4 Tutor-marked assignments (TMAs)
Examination

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.

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.

Entry requirements

You must declare the MSc in Mathematics (or another qualification towards which the module can count) as your qualification intention.

You should normally have a minimum of a 2:2 honours degree in mathematics or a 2:1 honours degree in a subject with a high mathematical content. If you don’t have such a qualification, your application will be considered and you may be asked to complete an entry test. Non-graduates will not normally be admitted to M823 unless as part of another Open University qualification. Students already registered for a qualification of which M823 is a constituent part will normally be admitted to M823.

You should have a good background in pure mathematics, with some experience in number theory and analysis. An adequate preparation would be our undergraduate-level modules Pure mathematics (M208) and Further pure mathematics (M303). A knowledge of complex analysis (as in, for example, Complex analysis (M337)) would be an advantage, but is not necessary. Note that if you wish later to study Analytic number theory II (M829), then knowledge of complex analysis is a requirement.

Whatever your background, you should assess your suitability for this MSc in Mathematics module by trying 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.

If you have any doubt about the suitability of the module, please speak to an adviser.

Register

Start End Fee Register
04 Oct 2025 Jun 2026 Not yet available

Registration closes 11/09/25 (places subject to availability)

Register
This module is expected to start for the last time in October 2027.

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.

Additional costs

Study costs

There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.

Study events

This module may have an optional in-person study event. We’ll let you know if this event will take place and any associated costs as soon as we can.

Ways to pay for this module

We know there’s a lot to think about when choosing to study, not least how much it’s going to cost and how you can pay.

That’s why we keep our fees as low as possible and offer a range of flexible payment and funding options, including a postgraduate loan, if you study this module as part of an eligible qualification. To find out more, see Fees and funding.

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 and tutor group 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.

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.

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.

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