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.
No current presentation
- see Future availability
This module is expected to start for the last time in October 2018.
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).
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 speak to an adviser.
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.
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.
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.
You will need a computer with internet access to study this module as the study materials and activities are accessible via a web browser. Any other computer-based activities you will need to carry out, such as word processing, using spreadsheets, taking part in online forums, and submitting files to the university for assessment, are specified in the module materials. If any additional software is needed for these tasks it will either be provided or is freely available.
We recommend either of the following:
Windows desktop or laptop computer running Windows 7 or later operating system
Macintosh desktop or laptop computer running OS X 10.7 or later operating system.
A netbook, tablet, smartphone or Linux computer that supports one of the browsers listed below may be suitable. The screen size should be at least 1024 (H) x 768 (W) pixels. If you intend to use one of these devices please ensure you have access to a suitable desktop or laptop computer in case you are unable to carry out all the module activities on your mobile device.
We recommend a minimum 1 Mbps internet connection and any of the following browsers:
Internet Explorer 9 and above
Apple Safari 7 and above
Google Chrome 31 and above
Mozilla Firefox 31 and above.
Note: using the latest version for your browser will maximise security when accessing the internet. Using company or library computers may prevent you accessing some internet materials or installing additional software.
See our Skills for OU study website for further information about computing skills for study and educational deals for buying Microsoft Office software.
Materials to buy
- 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.
Note: Presenting in October 2014 and October 2016 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 us if you want to know more about study with The Open University before you register.
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:
This module is only available in alternate even-numbered years. The details given here are for the module starting in October 2016.
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.
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.