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# Analytic number theory II

Analytic number theory is a vibrant branch of mathematics concerned with the application of techniques from analysis to solve problems in number theory. In this intermediate-level module, which is a sequel to Analytic number theory I (M823), you’ll learn about a rich collection of analytic tools that can be used to prove important results such as the prime number theorem. You’ll also be introduced to the Riemann hypothesis, one of the most famous unsolved problems in mathematics. Before embarking on M829, you should have completed a module in complex analysis, covering topics such as the calculus of residues and contour integration.

## Qualifications

M829 is an optional module in our:

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

## Module

Module code
M829
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
To enable you to make international comparisons, the information provided shows how OU postgraduate modules correspond to the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ).
Study method
Distance learning
Find out more in Why the OU?
Module cost
See Module registration
Entry requirements

Find out more about entry requirements.

## 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?

This module (and the preceding module Analytic number theory I (M823)) are about the application of techniques from analysis in solving problems from number theory. In particular, you'll learn about the prime number theorem, which estimates how many prime numbers there are less than any given positive integer. You'll also find out about the Riemann hypothesis, one of the most famous unsolved problems in mathematics. To understand these topics, you'll study certain rich classes of functions that are analytic in parts of the complex plane, among them the Riemann zeta function, which is the subject of the Riemann hypothesis.

This module is based on Chapters 8-14 of the set book Introduction to Analytic Number Theory by T. M. Apostol.

### 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.

## Teaching and assessment

• Marking your assignments (TMAs) and providing detailed feedback for you to improve.
• Guiding you to additional learning resources.
• Providing individual guidance, whether that’s for general study skills or specific module content.
• Facilitating online discussions between your fellow students, in the dedicated module and tutor group forums.

Module tutors also run online tutorials throughout the module. Where possible, recordings of online tutorials will be made available to students. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part. If you want to participate, you’ll likely need a headset with a microphone.

### 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 the 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

## Future availability

Analytic number theory II (M829) starts every other year – in October.

We expect it to start for the last time in October 2030.

## Regulations

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.

## Entry requirements

You must declare the MSc in Mathematics (or another qualification towards which the module can count) as your qualification intention. You must also:

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.

## Register

Start End Fee
- - -

No current presentation - see Future availability

This module is expected to start for the last time in October 2030.

## Future availability

Analytic number theory II (M829) starts every other year – in October.

We expect it to start for the last time in October 2030.

### Study costs

There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.

### Study weekend

This module normally includes an optional study weekend, for which there is an additional cost. You’ll pay this charge when you book, after you’ve registered on the module. If the study weekend is face-to-face, you must also pay for your travel to and from the venue and your accommodation if you need it.

## Study materials

### What's included

• a week-by-week study planner
• course-specific module materials
• audio and video content
• a specimen exam paper with solutions
• assessment details and submission section
• online tutorial access

You'll also be provided with printed course notes, which includes a narrative to accompany the module text, additional exercises and solutions.

### 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 an up-to-date version of Windows or macOS.

The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

To join in the spoken conversation in our online rooms we recommend a headset (headphones or earphones with an integrated microphone).

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.