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Analytic number theory I

Number theory has its roots in ancient history but particularly since the seventeenth century, it has undergone intensive development using ideas from many branches of mathematics. In spite of 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, in particular, 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 optional module in our:

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

Postgraduate Loans

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

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
Find out more in Why the OU?
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?

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

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

See the satisfaction survey results for this course.

Future availability

Analytic number theory I (M823) starts once a 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 2021.

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. 

    We recommend that you have at least second-class honours in mathematics. In exceptional circumstances applicants without this qualification will be considered, although non-graduates will not normally be admitted to the MSc programme. 

    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 England fee Register
    06 Oct 2018 Jun 2019 £1020.00

    New to OU? Registration closed but there may be places available. Ring +44 (0)300 303 5303. Already an OU Student? You have until 20/09/18 to register.

    Register
    This module is expected to start for the last time in October 2021.

    Future availability

    Analytic number theory I (M823) starts once a 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 2021.

    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.

    Ways to pay

    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. To find out more see Fees and funding.

    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.