- Points: 30
- Code: M823
- Level: Postgraduate

- On this page

A great module! I was a little daunted by the title of this unit - but I found it much more accessible than I was expecting. The textbook is excellent, the course notes are really helpful and if you get stuck the etutorials are always an option. This module fits very nicely with Analytical number theory II - and when you finish both you have a very good overview of the development of tools to help analyse the prime number theorem and the Riemann hypothesis. Highly recommended!

Andrew Chambers

Course starting: **October 2013**

Review posted: **June 2015**

I chose to start my MSc Maths journey with M823 because I had passed the (now defunct) module M381 Number Theory & Math logic at L3 and enjoyed the NT part. I did not feel 'brave' enough to attempt M820 Calculus of Variations without having done MS324 Waves, Diffusion & Variations.

You must purchase the set textbook (Apostol) for yourself but you are provided with OU course notes, giving explanations for the textbook and problems to do. TMAs, errata, etc are downloaded from the course webpage.

I found the module very hard - I was expecting a tough year and got one! Having got used to the very clear OU undergraduate maths texts, studying from a proper textbook came as a shock. I found it hard to make progress initially, but persevered, and the module started gradually to make more sense.

I found the subject matter deeply interesting, despite the hard grind, and I felt that I got a much deeper insight into the real number line and it's primes..

The TMAs were an order of magnitude tougher than those at L3. I found each one would take me about 3/4 weeks to do and I felt under significant pressure, especially as the exam approached. Be advised that the final TMA is regarded as a 'revision' one and contains an essay question...

My tutor, however, was excellent and the the online tutorials saved a long journey. At L8, you are only provided with 1 specimen paper to revise with. This was another shock as I had got used to purchasing lots of past papers as an undergraduate. However, our tutor provided extra questions. I found that revision was much harder than before, however.

I took the exam in June 2014. I managed to pass the module, enabling me to start the follow-up M829 Analytic Number Theory part ii the following October.

To sum up; very hard work, but a deeply interesting subject. I would advise anyone considering this to expect a very different experience from undergraduate OU maths. Do not think that you will 'waltz' through this 'entry level' module just because you have a degree in Maths!

Christopher Bernard Clarke

Course starting: **October 2013**

Review posted: **June 2015**

I did M381 and obtained distinction, so thought this would be easy. But most of it was very different, so it took me some time to understand what it was all about.

But, after I had read the notes and done the exercises for a section, I tried the TMA question for that section. This has always worked for me. So, it got easier and easier. The chair sent us some specimen questions not long before the exam, which turned out to be very similar to the exam. I Got another distinction at the age of 75. Do this course.

Vincent James Lynch

Course starting: **February 2013**

Review posted: **June 2014**

I had bought the set book several years before, when Amazon had a special offer on it. I was doing number theory and mathematical logic at the time. I found the book very difficult at that time. I would still find it difficult to reproduce proofs printed in this book, but you are not asked to do that in the exam.

The e-tutorials were very useful, and often showed techniques which were useful in solving TMA problems but were not explicitly exampled in the course notes. I probably spent more time on this than any other module I have ever done, but my final revision came when the e-tutorial tutor sent us another specimen paper. That enabled me to get a Distinction.

If I compare it to M820, the course I am doing now, it needs e-lectures. I have found these very helpful.

Vincent James Lynch

Course starting: **February 2013**

Review posted: **February 2014**

M823 is a relatively straightforward course in that it follows the first seven chapters of Apostol's "Introduction to Analytic Number Theory". This is a rigorous but often terse book which I liked but many others on the course seemed to find difficult. The course as presented is slightly light in content so Chapter 9 of Apostol on quadratic reciprocity is added to pad it out.

M829 completes the "story" of the Prime Number Theorem but, for me, getting to Dirichlet's Theorem on primes in arithmetic progressions at the end of chapter 7 was sufficient.

The course was very interesting and contains a fascinating insight into the mathematics required to attack the Prime Number Theorem. There is a significant amount of technical detail and the main thing that I learned was how calculus can be applied to number theory.

Assessment of the course is difficult for the question setters - there appear to be only a limited number of question types that can reasonably be asked - which of course is a bonus for the hardworking student!

The course was well supported by very enthusiastic tutors.

Course starting: **February 2013**

Review posted: **December 2013**

M823 is an interesting course bringing together analytic methods and number theory practices. Much of the course consists of methods which border on elementary number theory and the most analysis is around the use of big and small O order arguments.

Despite this, the course has a strong rigorous base and the textbook by Tom Apostol is well-written and very coherent. Some found his style off-putting but I found it very engaging.

The TMAs are tricky as opposed to difficult I think. However, the exam that I sat was similar to the specimen and didnt include many difficult tricks; - as such, this course provides an excellent entry into Analytic Number Theory.

Most people will find themselves subsequently drawn to M829 which completes the book. Overall, I didnt expect to like this course as I never liked number theory at UG but found it to be very enjoyable and quite easy-going for the main part. You could certainly do this alongside another module.

Course starting: **February 2012**

Review posted: **October 2013**

E-tutorials saved me having to pay for travel, also, would've struggled with the set course book without the additional material. I didn't do well enough to do analytic number theory II, but the course is absolutely brilliant. The first 3 TMAS don't count so you get a 'feel' for what is required without being penalised(although the substitution mark scheme on other courses does take this into account).

I didn't get a first class maths degree so i'm pleased I got through and felt very good being with the other students! Lol, I can't call myself an amateur number theorist with my percentage but I tried!

The big bonus is the cost, for this level of study it's very cheap!

Adam Bell

Course starting: **February 2012**

Review posted: **December 2012**

An extremely well structured and thoroughly interesting course. Takes off where elementary number theory M831 leaves off. This is a very fundamental area of pure mathematics with a very profound and interesting history involving some of the really great mathematical minds of the last few hundred years(Gauss, Fermat, Legendre, Abel ,...etc). It is still a highly active area of research and in this course some of the excitement of the uncharted territory is conveyed.

The set book Apostol is a classic text and exceptionally well laid out. The course notes are very clear and helpful with for the most part well explained solutions. The TMAs take a fair bit of thinking and time to do well and are definitely a significant step up from the undergraduate courses and the exam likewise is very time pressured with quite challenging questions mixed with some more manageable ones.

Conceptually there is a lot of very intellectually stimulating material in this course and some of the analytic methods are really impressive as tools particularly at studying the distribution of primes. In fact there is little if anything other than positive remarks I can make regarding this course. I would recommend it to anyone with a a genuine love of maths ( but this is a pure course which has applications in cryptology and other fields no applications are discussed in this course).

Full marks to the course organisers for a first class course. As for me - I am starting the follow up M829. Nuff said.

Course starting: **February 2008**

Review posted: **February 2009**

I enjoyed this course very much, the set book and the course notes are both excellent.

The sample exam was a fair reflection of the actual exam, though I found myself a bit lost for time to do all the required questions and now know I should have had more exam practice.

Word of warning though! Give yourself some extra time for chapters 3,4 and 7. And you will probably need to review/study basic Elementary Number Theory (though they did publish, on the course website, the OU books from the undergraduate course)

I highly recommend this course to anyone, though it is definitely one of the more Pure subjects, and this might not suit the more Applied minded, but there are some very interesting results produced.

Course starting: **February 2008**

Review posted: **December 2008**

This was my first MSc course, and I thoroughly enjoyed it. The set book by Apostol is very well written, and the course notes provide extra help where needed. I'd done some number theory before, but this isn't essential as the book covers all the necessary background.

A couple of the more technical chapters were a bit daunting at first, but the exercises give plenty of practice in the key results and techniques. Overall I found the study time required was similar to 3rd level maths courses, but the TMAs needed more thinking time. The exam had a good range of questions to choose from; the main challenge was the time pressure.

Course starting: **February 2005**

Review posted: **April 2006**

I had not done any Number Theory, so at first sight, this course looked daunting. However, the course notes are very good, and if you do all the SAQs, you get plenty of practice for the questions in the TMAs.

I found the exam a bit long, but covered all the course, so no real surprises. Overall an excellent course and a great grounding for other Number Theory areas (e.g. Riemann's Hypothesis)

Dugald Henderson

Course starting: **February 2005**

Review posted: **January 2006**

Each of the views expressed above is an individual's very particular response, largely unedited, and should be viewed with that in mind. Since modules are subject to regular updating, some of the issues identified may have already been addressed. In some instances the faculty may have provided a response to a comment. If you have a query about a particular module, please contact your Regional Centre.

The figures below are taken from a survey of students who sat the exam/completed the end-of-module assessment for the October 2016 presentation of M823. The survey was carried out in 2017.

34 students (a response rate of 41%) responded to the survey covering what they thought of 10 aspects of the module.

*Please note that if the percentage of students who responded to this module survey is below 30% and/or the number of responses is below 23 it means that only a small proportion of students provided feedback and their views as shown here may not be fully representative of all students who studied the module.*

See this page for the full text of questions and more information about the survey.

% | Count | |
---|---|---|

Overall, I am satisfied with the quality of this module | 91.2 | 31 |

Overall, I am satisfied with my study experience | 88.2 | 30 |

The module provided good value for money | 57.6 | 19 |

I was satisfied with the support provided by my tutor/study adviser on this module | 94.1 | 32 |

Overall, I was satisfied with the teaching materials provided on this module | 82.4 | 28 |

The learning outcomes of the module were clearly stated | 85.3 | 29 |

I would recommend this module to other students | 85.3 | 29 |

The module met my expectations | 88.2 | 30 |

I enjoyed studying this module | 88.2 | 30 |

Overall, I was able to keep up with the workload on this module | 91.2 | 31 |

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Faculty comment:"The module team was pleased that the overall levels of satisfaction demonstrated by the survey results were high in 2016. Once again, the online tutorials and screencasts were particularly appreciated, and the module team will make every effort to ensure that these and other aspects beyond the core text give students a sense of 'value for money'."