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Accessibility statement
A image to illustrate Analytic number theory II module
Number theory has roots in ancient history, but since the seventeenth century, it’s developed intensively using ideas from many branches of mathematics. Despite 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, you’ll study number theory using techniques from analysis, particularly the convergence of series and integration methods. Based on readings from T.M. Apostol’s Introduction to Analytic Number Theory, the module builds toward understanding and proving Dirichlet’s theorem on the distribution of primes in arithmetic progressions.
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:
Analytic number theory is about applying techniques from analysis to tackle problems in number theory. In this module, which is a sequel to Analytic number theory I (M823), you’ll learn about Gauss sums, primitive roots, Dirichlet series, Euler products, Dirichlet L-functions, the gamma function and the Riemann zeta function. Highlights of the module include a proof of the celebrated prime number theorem and an introduction to the Riemann hypothesis, one of the greatest unsolved problems in mathematics. The module finishes with applications of analytic number theory to the subject of integer partitions.
This module is based on Chapters 8-14 of the set book Introduction to Analytic Number Theory by T. M. Apostol.
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.
You’ll get help and support from an assigned tutor throughout your module.
They’ll help by:
Online tutorials run throughout the module. While they’re not compulsory, we strongly encourage you to participate. Where possible, we’ll make recordings available.
Course work includes:
We regularly review the assessments in our modules, so we may update the examination method used for this module to an in-person exam or a remotely invigilated (proctored) exam. When we are making such a change, we will make it clear on this page. If we need to make a change after you have registered, we will notify you as soon as possible. If you have additional requirements, we will support you to complete your assessments.
You’ll have access to a module website, which includes:
You’ll also receive printed course notes, which include a narrative to accompany the module text, additional exercises, and solutions.
Set books:
You can study this module on its own or use the credits you gain towards an Open University qualification.
M829 is an option module in our:
Analytic number theory II (M829) starts every other year – in October.
It will next start, for the final time, in October 2026.
As a student of The Open University, you should be aware of the content of the academic regulations, which are available on our Student Policies and Regulations website.
You must have passed:
You should have studied an undergraduate course in complex analysis covering topics such as the calculus of residues and contour integration. Complex analysis (M337) should provide adequate preparation.
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.
To find out more about what kind of support and adjustments might be available, contact us or visit our disability support pages.
StartEndRegister byEngland fee
03 Oct 202630 Jun 202710 Sep 2026£1,550
This module may have an optional study weekend. There will be a charge to attend in person or online. If attending in person, you’ll also need to pay for your travel, and accommodation if you need it. More details will be available after the module has started.
If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to support your study costs. To find out more, see Postgraduate loans in England.
Studying with The Open University can boost your employability. OU courses are recognised and respected by employers for their excellence and the commitment they take to complete. They also value the skills that students learn and can apply in the workplace.
Over 30,000 employers have used the OU to develop staff so far. If the module you’ve chosen is geared towards your job or developing your career, you could approach your employer to see if they will sponsor you by paying some or all of the fees.
You can pay part or all of your tuition fees upfront with a debit or credit card when you register for each module.
We accept American Express, Mastercard, Visa and Visa Electron.
Please note: your permanent address/domicile will affect your fee status and, therefore, the fees you are charged and any financial support available to you. The fee information provided here is valid for modules starting before 31 July 2027. Fees typically increase annually. For further information about the University's fee policy, visit our Fee Rules.
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