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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:

Postgraduate Loans 

If you study this module as part of an eligible qualification, you may also be eligible for a Postgraduate Loan available from Student Finance England. For more information, see Fees and funding.

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
Postgraduate
Study method
Distance learning
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Module cost
See Module registration
Entry requirements

Find out more about entry requirements.

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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

You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance.

Contact us if you want to know more about study with The Open University before you register.

Assessment

The assessment details can be found in the facts box above.

You will be expected to submit your tutor-marked assignments (TMAs) online using eTMA system, unless there are some difficulties which prevent you from doing so. In these circumstances, you must negotiate with your tutor to get their agreement to submit your assignment on paper.

Course work includes

4 Tutor-marked assignments (TMAs)
Examination
No residential school

Course satisfaction survey

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Future availability

Analytic number theory II (M829) starts every other year – in October. This page describes the module that will start in October 2018. We expect it to start for the last time in October 2022.

Regulations

As a student of The Open University, you should be aware of the content of the academic regulations which are available on our Essential Documents 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 England fee Register
    06 Oct 2018 Jun 2019 £1020.00

    Registration closes 16/08/18 (places subject to availability)

    Register
    This module is expected to start for the last time in October 2022.

    Future availability

    Analytic number theory II (M829) starts every other year – in October. This page describes the module that will start in October 2018. We expect it to start for the last time in October 2022.

    Additional costs

    Study costs

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

    Study weekend

    This module normally includes an optional study weekend. For each day you choose to attend, you must pay an additional charge of around £50 to cover tuition and refreshments during the day. You’ll pay this charge when you book, after you’ve registered on the module. You’ll also have to pay for your own travel to and from the venues and your own accommodation if you need it.

    Study materials

    What's included

    Module notes, other printed materials.

    You will need

    We recommend that you have access to the internet at least once a week during the module and would like to point out that vital material, such as your assignments, will be delivered online.

    Computing requirements

    A computing device with a browser and broadband internet access is required for this module.  Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.

    Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

    A desktop or laptop computer with either:

    • Windows 7 or higher
    • macOS 10.7 or higher

    The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

    To participate in our online-discussion area you will need both a microphone and speakers/headphones. 

    Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students. 

    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. Find out more about our services for disabled students.