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Analytic number theory II

Analytic number theory is a vibrant branch of mathematics concerned with the application of techniques from analysis to solve problems in number theory. In this intermediate-level module, which is a sequel to Analytic number theory I (M823), you’ll learn about a rich collection of analytic tools that can be used to prove important results such as the prime number theorem. You’ll also be introduced to the Riemann hypothesis, one of the most famous unsolved problems in mathematics. Before embarking on M829, you should have completed a module in complex analysis, covering topics such as the calculus of residues and contour integration.

Qualifications

M829 is an optional module in our:

This module can also count towards M03, which is no longer available to new students.

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

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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?

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
No residential school

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.

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 must also:

    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
    - - -

    No current presentation - see Future availability
    This module is expected to start for the last time in October 2030.

    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.

    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 has an optional study weekend. You must pay £60 for tuition and refreshments. You must also pay for your travel to and from the venue, and accommodation if you need it.

    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 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 (11 'Big Sur' 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.

    If you have particular study requirements please tell us as soon as possible, as some of our support services may take several weeks to arrange. Visit our Disability support website to find more about what we offer.