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Analytic number theory II

Analytic number theory is a vibrant branch of mathematics concerned with applying techniques from analysis to solve number theory problems. This module follows on from the module Analytic number theory 1 (M823). You’ll learn about a rich collection of analytic tools that can prove results, such as the prime number theorem. This module also introduces the Riemann hypothesis, one of mathematics’s most famous unsolved problems.

Qualifications

M829 is an option module in our:

Module

Module code
M829
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 n2 + 1?

  • Does there always exist a prime between n2 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.

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 II (M829) starts every other year – in October.

This page describes the module that will start for the final time in October 2026.

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 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.

Register

Start End Fee
- - -

No current presentation - see Future availability
This module is expected to start for the last time in October 2026.

Future availability

Analytic number theory II (M829) starts every other year – in October.

This page describes the module that will start for the final time in October 2026.

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 weekend

This module may have an optional study weekend; there will be a charge to attend in person or online. If attending in person, you must also pay for travel and accommodation if you need it. More details will be available after the module has started.

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 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.

Computing requirements

  • Primary device – A desktop or laptop computer. It’s possible to access some materials on a mobile phone, tablet or Chromebook; however, they will not be suitable as your primary device.
  • Peripheral device – Headphones/earphones with a built-in microphone for online tutorials.
  • Our OU Study app operates on supported versions of Android and iOS.
  • Operating systems – Windows 11 or latest supported macOS. Microsoft will no longer support Windows 10 as of 14 October 2025.
  • Internet access – Broadband or mobile connection.
  • Browser – Google Chrome and Microsoft Edge are recommended. Mozilla Firefox and Safari may be suitable.

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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