Analytic number theory I
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Summary
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, many problems that are easy to state and understand still exist – for example, is there an even number > 2 that is not the sum of two primes? In this course (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 course is based on readings from T. M. Apostol’s Introduction to Analytic Number Theory.
Course content
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 250 AD the subject stagnated until the seventeenth century. Since then there has been intensive development, using ideas from many branches of mathematics. The field of number theory is now vast and many of the problems in it are difficult to understand but in spite of the maturity of the subject, many problems that are easy to state and understand still exist – for example:
- Is there an even number (> 2) that is not 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 n2 + 1?
- Does there always exist a prime between n2 and (n + 1)2 for every integer n > 1?
In this MSc course and in Analytic number theory II (M829), we study number theory using techniques from analysis in particular, the convergence of series and the calculus of residues. Among the results proved in M823 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.
M823 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.
Entry
You must be registered for the MSc in Mathematics, or for another qualification towards which the course can count. You should have at least second-class honours in a mathematics degree or in another degree with a high mathematical content, such as engineering or theoretical physics. 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 taken good first courses in real analysis and complex analysis. An adequate preparation would be our undergraduate-level courses Pure mathematics (M208) (or the discontinued course M203) and Complex analysis (M337) (or the discontinued courses M231 and M332). A knowledge of elementary number theory (as in, for example, Number theory and mathematical logic (M381) or the discontinued courses M382 or M383) would be an advantage, but is not necessary. If you have any doubt about the suitability of the course, please contact our Student Registration & Enquiry Service.
Qualifications
M823 is an optional course in our
- Postgraduate Certificate in Mathematics (C90)
- Postgraduate Diploma in Mathematics (E23)
- MSc in Mathematics (F04)
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 course can count towards these qualifications because from time to time the structure and requirements may change.
If you have a disability or additional requirement
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 are a new student, or new to courses using a computer or the internet, you will need to inform us of your particular needs as soon as possible, as some of our support services may take several weeks to arrange. Details of how to do this and our range of support services are described in our booklet Meeting Your Needs which you can download or request from our Student Registration & Enquiry Service.
You can also find information about accessible course materials, financial support and the Disabled Students' Allowance, equipment and other services, on our Services for disabled students website. It also includes our contact details for advice and support both before you register and while you are studying.
Course materials
What's included
Course notes, other printed materials.
You will need
We recommend that you have access to the internet at least once a week during the course and would like to point out that vital material, such as your assignments, may be delivered online.
Computing requirements
This course has online computer activities – you can access these using a web browser that can play Flash and Shockwave.
You will need internet access and a computer. If you have purchased a new computer since 2002 it should meet your course computing requirements. Check our Technical Requirements section if your computer is older than this or is otherwise unusual.
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 and also Eddington Hook, tel. 0800 018 2799, www.eddington-hook.co.uk/ou, who may have copies in stock.)
Teaching and assessment
Support from your tutor
You will have a tutor who will help you with the course 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 for this course can be found in the facts box above.
TMAs on MSc in Mathematics courses can be submitted electronically using a special maths eTMA processor which is used in place of the main eTMA system. We strongly recommend you submit the TMAs electronically, but there will be the facility to submit TMAs on paper.
Assessment is an essential part of the teaching, so you are expected to complete it all. But if you unavoidably miss or do badly in an assignment, some courses allow you a ‘substitution score’, calculated as a weighted average of all your scores for the course. In M823 this rule can apply to one assignment only. You will be given more detailed information when you begin the course.
Students also studied
Students who studied this course also studied at some time:
Future availability
The details given here are for the course that starts in February 2010. We expect it to be available once a year.
How to register
Unfortunately, we are unable to accept online registrations for this course in the usual way. Instead, you may apply electronically using the MSc Mathematics application system.
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A postgraduate course in Mathematics and Statistics.
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Student Reviews
An extremely well structured and thoroughly interesting course. Takes off where elementary number theory M831 leaves off. This is a ...
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I enjoyed this course very much, the set book and the course notes are both excellent. The sample exam was ...
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