As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

Please follow this blog

Search this blog

Thursday, March 10, 2011

PNT is the most difficult theorem I encountered thus far

$$\sum_{d/n}\frac{\mu(d)}{d} = 1 -\sum \frac{1}{p_i} + \sum \frac{1}{p_i p_j} - \sum \frac{1}{p_i p_j p_k} + \cdots$$
Apostol, Chapter 2: Arithmetical Functions

Although I am doing M381 and M373 this year, my real goal for this year is ( read: was ) self-studying Apostol's Analytic Number Theory. I came to realize that a goal like that must be wrong if I want to go anywhere. It may look like a SMART goal but it is not. ( SMART goals are S(pecific), M(easurable), A(chievable), R(ealistic) and T(rackable). ) Since self-study projects are not finished with an exam, it is hard to measure if the goal has been achieved, the goal is not specific enough. After about 1/4th of the study year I changed my goal to: "understanding the proof of the prime number theorem (PNT)". Although the main purpose of Apostol's textbook is just that. I am now free of the strict planning I imposed on myself, i.e. March: do AANT Chapters 2 and 3, etc. I can focus entirely on understanding the proof no matter where the knowledge comes from. Once I understand the theorem I should have a new outlook on the Number Theory field and decide where to go from there.

I chose Apostol's book because it is used in OU Courses M823 and M829, of which I sincerely hope are still around when I am eligible to do them. I thought it was more or less the Analytical Number Theory Bible, well it is not, it seems. The classic work in Analytical Number Theory is the following book by Iwaniec and Kowalski.


  1. Nilo your ability to find out what books are relevant and are not is really amasing. I also concur with your view that it is better to concentrate on what you need to know to understand a theorem or calculation rather than say 'working through' books systematically. I guess it's a sign of maturity perhaps you no longer need the crutch of OU maths to pursue your dream. Especially if fees rocket through the roof as they will do sooner or later

    Best wishes Chris

  2. Hi Chris,

    You wrote: "... perhaps you no longer need the crutch of OU maths to pursue your dream ..." I do, at least to justify my behavior. Very few people understand the thrill of mathematics.

  3. Yes me to at least till I finish the degree the MSc. Unfortunately you may end up doing courses just to fill in the gaps as I feel I'm doing with Linear Statistical modelling. Also full qualifications are your entry ticket into research. The problem there is of course even if one gets distinction for both the Degree and the MSc it's no guarantee of a place and the opportunities to do part time PHd's in maths by say distance learning are non existent at least here in Britain. When the fees go up I would probably have to do only one module a year which means that it would take 6 long years to get the MSc. As I already have the books and the course notes for most of the modules I really do have to question whether or not it's worth it. Still I wont be making up my mind just yet.

  4. Chris,
    I really hope we meet someday. This year, perhaps as I could do tutorials in Edinburgh. I sincerely think your possibilities are greater than you give yourself credit for. Society is programmed to make people behave like ants. Ants don't do mathematics.

    I think you are going through the equivalent of my "Why am I doing this?"-MT365 experience with LSM.

    Don't forget your goal: understanding ( improving on ) the Big Bang quantum thing.

    I think I pay three times as much as you do. It is part of the reason why I do 60 this year. Although I somewhat regret I didn't do Complex Analysis as well. - I may have a few goodies for you. ;-)


  5. I found "Problems in Analytic Number Theory" to be a good complement to the Apostol.

  6. Dranco, Thanks.
    I have the book now. It is a beautiful book indeed. Geared for 'study by doing' very applicable to number theory.


Popular Posts

Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)