Re: Number Theory - Study methods
- From: Angus Rodgers <twirlip@xxxxxxxxxxx>
- Date: Sun, 28 Oct 2007 18:08:25 +0000
On Sun, 28 Oct 2007 10:31:50 -0000, Jeremy Boden
<jeremy@xxxxxxxxxxxxxxx> wrote:
I will be returning to University level study soon.
This is a distance learning taught Masters degree - starting with Number
Theory - based on extracts from the first 6 chapters of Apostol's
Analytic Number Theory (which I already have).
It's quite a few years since I last did proper directed study and I am a
little unsure about my ability to cope with this course. So I suppose I'm
looking for advice on study methods.
Since I have 3 months before the course starts, can anyone make
*constructive* suggestions as to how I should use my time?
I find that "read the book" is not particularly helpful.
This is an Open University course and they offer lot's of advice on study
methods - unfortunately all related to their Arts courses..
I don't know if there are any "methods", and studying is a
very individual matter. I remember thinking that C. A. Mace's
Pelican paperback "The Psychology of Study" was quite helpful,
but it's years since I read it. What I mainly remember from
it is that you should keep your own questions in mind, as your
own individual motivation for studying.
One "method" I do use, which might just seem odd to anyone
else, but which gives me confidence (as someone else who has
recently returned to formal study of maths with the OU), is to:
(i) Keep all my notes (unless of obviously only very temporary
interest - these can be consigned to any old scraps of paper)
in neat little A5 notebooks (the ones I've been using for my
OU courses no longer seem to be easily available, but I found
some in Woolworths which should do very well). I make no
attempt to write them up in final form, and I don't worry
about how stupid they are; they are an honest record of my
thought processes, and valuable as such.
(ii) Although I often have to refer back to these rough notes,
for unexpected reasons, I don't like to have to do so too often,
so I use LaTeX to type up anything I think I am going to have to
refer to later. LaTeX is also a very good typographic tool for
writing your TMAs with. (I also use it for everyday letters, and
never touch a word processor.) It gives me a silly, narcissistic,
but undeniably real sense of confidence to see that even my own
often muddled crap can be typed up to look as neat as anything
in a textbook. Try it! You'll soon be addicted. :-)
(iii) Use the OU FirstClass conferences (general and course-
specific) to share ideas, to keep yourself sane (if necessary),
and to stop yourself from slacking off (also if necessary).
Another general bit of advice that works for me is "A change
is as good as a rest." That is, I find it helpful to study
several aspects of mathematics at the same time: alternating
between them, of course, and preferably spending at least an
hour at a time on each topic, so that I have time to warm up
and really get into the subject.
(But I must admit, I've been slacking off badly recently!)
Regarding the course M823 Analytic Number Theory I itself:
(iv) It would probably be useful to know something about
arithmetical functions ahead of the course. It's not that
they're not covered thoroughly in Apostol and in the course
notes - they are - but that there is a lot to remember, and
I (for one) found it hard to keep it all in mind. However,
the exam is an "open book" one, so you won't have to depend
on having a good memory. (Mine is absolutely terrible.)
(v) When it gets near exam time, it would be a good idea to
practice (e.g. on the specimen paper) working against the
clock. You have to answer 9 questions in 3 hours (unless
they've changed the rubric since 2005), and although I knew
how to do all the questions (I seem to remember the specimen
paper is good preparation - unfortunately, due to domestic
pressures, I never got round to looking at it until the day
of the exam!), I ran out of time. Other students have also
commented that the exam is a test of speed. (I used to be
fast when I was young, but I'm not now!)
(vi) You won't need to know any complex analysis until M829
Analytic Number Theory II. (It's fortunate for me, as I still
hardly know any!) But you will need to be familiar with the
most elementary facts about complex numbers for the section
on group characters. It would also be an advantage to be
familiar with some of the (very) elementary theory of (finite,
abelian) groups (but not even including the structure theorem).
I wouldn't like to have been meeting that for the first time
when I was doing the course.
(vii) Probably the main prerequisite is a solid understanding
of basic real analysis. I don't think a lot of deep theorems
are involved, just a lot of manipulation.
(viii) One thing, however, that I did find very challenging
was understanding Apostol's use of the O() notation. I won't
rehash my long wrangle with David Ullrich over this (please!);
I'll just say that you need to pay attention to the explicit
lower bound on the values of x for which an inequality like:
|f(x)| <= Mg(x) (x >= a)
is true. Apostol rather gives the impression that the lower
bound on x (i.e. the number a, where |f(x)| <= Mg(x) for all
x >= a) is existentially quantified, i.e. the statement "f(x)
= O(g(x))" means that the stated inequality holds for /some/
value of a, which is not needed again and can be forgotten.
In fact sometimes a has to be equal to 1 (and Apostol does
prove all the relevant inequalities for the right value of a);
also, if I remember correctly, sometimes Apostol writes in
effect "for all x > 1" when he should write "for all x >= a,
for some a > 1". My notes tell me that Lemma 7.5 is a case
in point. I don't want to alarm you (still less do I want to
provoke David Ullrich again!), and Apostol's arguments are
all perfectly valid so long as you keep this caution in mind.
I mention it because it's not mentioned in the course notes,
and it caused me some genuine difficulty (and much confusing
argument!).
Any references to non-spammed news groups would be of interest too.
I'm not aware of an alternative to sci.math. It is a very
active group, and there are a lot of cranky threads (/pace/
galathaea!), but a killfile takes care of a lot of that. I
use the "German news server" <http://news.individual.net/>,
which perhaps filters out a lot of actual spam, because I
don't see much - what do you consider "spam"? - unless you
mean the constant infuriating requests for solution manuals,
which is a relatively recent curse to have descended on the
group. (May it pass soon.)
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.
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