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Analytic number theory II

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 intermediate-level module (and in Analytic number theory I (M823)), 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.

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No current presentation - see Future availability

This module is expected to start for the last time in October 2016.

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 intermediate-level module (and in Analytic number theory I (M823)), 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 is the prime number theorem, which estimates the number of primes up to a given value x.

This module is based on Chapters 8-14 of the set book Introduction to Analytic Number Theory by T. M. Apostol (1986, fourth edition, Springer-Verlag).

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 is one of the intermediate modules in the MSc in Mathematics (F04). Normally, you should have completed at least one intermediate module before studying Advanced mathematical methods (M833).

Entry

To study this module you must:

  • declare the MSc in Mathematics (or another qualification towards which the module can count) as your qualification intention
  • have successfully completed Analytic number theory I (M823)

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 contact our Student Registration & Enquiry Service.

Qualifications

M829 is an optional module in our:

Some postgraduate qualifications allow study to be chosen from other subject areas. We advise you to refer to the relevant qualification descriptions for information on the circumstances in which this module can count towards these qualifications because from time to time the structure and requirements may change.

Regulations

As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are available on our Essential documents website.

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.

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

You will need a computer with internet access to study this module as it includes online activities, which you can access using a web browser.

  • If you have purchased a new desktop or laptop computer since 2008 you should have no problems completing the online activities.
  • If you’ve got a netbook, tablet or other mobile device check our Technical requirements section.
  • If you use an Apple Mac you will need OS X 10.7 or later.

You can also visit the Technical requirements section for further computing information (including details of the support we provide).

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

Presented October 14 and October 16 only

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 our Student Registration & Enquiry Service 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 a special maths eTMA processor, which is used in place of the main 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.

You will, however, be granted the option of submitting on paper if typesetting electronically or merging scanned images of your answers to produce an electronic TMA would take you an unacceptably long time.

Students also studied

Students who studied this course also studied at some time:

Future availability

The details given here are for the module starting in October 2014. We expect it to be available in alternate years after that, in October.

How to register

We regret that we are currently unable to accept registrations for this course. Where the course is to be presented again in the future, relevant registration information will be displayed on this page as soon as it becomes available.

Distance learning

The Open University is the world's leading provider of flexible, high quality distance learning. Unlike other universities we are not campus based. You will study in a flexible way that works for you whether you're at home, at work or on the move. As an OU student you'll be supported throughout your studies - your tutor or study adviser will guide and advise you, offer detailed feedback on your assignments, and help with any study issues. Tuition might be in face-to-face groups, via online tutorials, or by phone.

For more information about distance learning at the OU read Study explained.

Course facts
About this course:
Course code M829
Credits 30
OU Level Postgraduate
SCQF level 11
FHEQ level 7
Course work includes:
4 Tutor-marked assignments (TMAs)
Examination
No residential school

Student Reviews

“A perfect conclusion to M823. Whilst the M823 stands well as a course in its own right I personally found...”
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“An extremely good follow up course to Analytic Number Theory I, and by the end of the course, I felt...”
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