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Student and tutor module reviews

Analytic number theory II

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  • Points: 30
  • Code: M829
  • Level: Postgraduate

Student reviews

This was possibly my favourite in the entire course. It builds very well from Analytic Number Theory I, and provides a very good conclusion to the content that you study in this first module. It's significantly harder than Analytic I - in particular there is quite a bit of complex analysis and complex integration in some chapters. However even though I didn't remember much of this I was able to get up to speed by downloading the OU's undergraduate on this and coped fine in the end. The first TMA was pretty daunting - but persevere!

My favourite part of the module was the content that built up to understanding the prime number theorem and the Riemann Hypothesis - and by the end of the course you have a good understanding of how these are derived. The Riemann Hypothesis is one of the most important unsolved mathematical problems which has withstood the attempts of some of the best mathematicians to crack it for over 100 years, so it was great to get to the stage of at least understanding the hypothesis during my MSc course!

The support from my tutor was great, there were lots of good screencasts online and both the set textbook and course notes were good.

The exam was pretty tough - quite a lot harder than the specimen paper, and there was a real time pressure. Still I think it was fair - after all you can't expect OU level 2 modules to be easy. This one isn't, but it is very rewarding.


Andrew Ian Chambers

Course starting: October 2014

Review posted: August 2015

A perfect conclusion to M823.
Whilst the M823 stands well as a course in its own right I personally found this follow up M829 essential in completing the picture: the analytic proof of the Prime Number Theorem (PNT) is the main objective. A significant amount of complex analysis is used to prove these results. The course gives a clear insight of the role of the Riemann zeta function in the proof of the PNT and a much deeper study of related Dirichlet series is also undertaken. Gauss and Ramanujan sums are explored and shown to provide an alternative proof of quadratic reciprocity. Other really interesting topics were Primitive roots, Hurwitz zeta functions and partitions. In summary the course is a thoroughly well presented follow up to M823 with a similar level of difficulty in TMA and exam. I would say if you like number theory then go for this one!

Course starting: February 2009

Review posted: June 2010

An extremely good follow up course to Analytic Number Theory I, and by the end of the course, I felt that the material had been wrapped up well and that I had gained significant knowledge of the subject.

My one criticism is that it can be a little difficult to complete this course if there is a long time gap between starting it and finishing Analytic Number Theory I. Of course, that's partly my fault - I should have taken this course immediately after. As it was, the first course was my first course on the Masters, and the second one was my last! I guess I can't really criticise this point too strongly.

Overall, though, a fine course. Nice.

Kin Ming Hung

Course starting: February 2007

Review posted: August 2008

If you are thinking of taking this course, you will already know that you must have taken M823 first and may therefore already be familiar with the set book, Apostol.

The distinguishing merit of these courses is that Apostol's presentation is admirably elegant and clear. There are no cul-de-sacs. Like musical motifs, ideas introduced in the early stages inevitably resurface to play their part in more advanced theorems. This seamless treatment means that the text requires only slight amplification in the course notes, which instead consist mostly of practice questions and a supplement on the prime number theorem (which *is* examined).

As the title suggests, the core of the course develops the intricate and fascinating interconnections between the arithmetical functions, transcendental functions (gamma and zeta) and Bernoulli polynomials, culminating in the classical proof of the prime number theorem. From complex analysis, you will need to dust off contour integration and the residue theorem but little more.

The remainder of the course (periodic functions, primitive roots and partitions) however accounts for half the examination questions and in my sitting these were more challenging than the course work might have led one to expect. Manipulating Ramanujan's sum and related tools does need practice and extra time honing your technique with the problems in Apostol ought to be rewarded.

In volume, if not in sophistication, M829 is comparable to an average undergraduate course. (My personal weighting is "60": see my reviews of M826 and M828.)

Course starting: February 2004

Review posted: September 2005

Please note

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.


Module satisfaction survey

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 M829. The survey was carried out in 2017.

17 students (a response rate of 34.7%) 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 94.1 16
Overall, I am satisfied with my study experience 94.1 16
The module provided good value for money 64.7 11
I was satisfied with the support provided by my tutor/study adviser on this module 76.5 13
Overall, I was satisfied with the teaching materials provided on this module 94.1 16
The learning outcomes of the module were clearly stated 100 17
I would recommend this module to other students 88.2 15
The module met my expectations 88.2 15
I enjoyed studying this module 88.2 15
Overall, I was able to keep up with the workload on this module 88.2 15
Faculty comment: "We thank students and tutors for making the module that started in October 2016 a success. We appreciate the positive feedback, and we'll use the constructive comments to improve the module and make it better value for money. In particular, we plan to create a series of screencasts on applications of complex analysis in analytic number theory. We'll also produce an enhanced, polished version of the revision exercise booklet to ensure that students are thoroughly prepared for the examination."
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