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    • CommentAuthorSirrush
    • CommentTimeJan 28th 2013

    I want to get an overview of the subject of irrational and transcendental numbers (most important ideas, methods and theorems - and especially what ideas are possible to push further).

    Is it ok to ask about that here?

    Here is what I know about so far, but I didn't study the details of it all yet:

    * logs (easy) and roots (Gauss)
    * existence of trancendental numbers: Cantor
    * construction: Liouville
    * e: series
    * pi: Lambert, Niven, Apery, Flint-Hills
    * e^pi: complex analysis
    * reciprocal fibonacci constant: André-Jeannin
    * Erdos-Borwein constant
    * Thue-Morse constant: Dekking
    * Euler: zeta(2n)
    * Apery: zeta(3)
    * Wadim Zudilin: zeta(5),zeta(7),zeta(9),zeta(11)

    * Dirichlet, Kronecker, Weyl equidistribution, Markoff-Lagrange spectrum
    * Mahler measure
    * Lindemann-Weierstrass, Gelfond-Schneier, Baker's linear forms in logarithms

    * pi + e
    * Catalans constant
    * Euler-Mascheroni

    The area known roughly as "transcendental number theory" is awfully big, and in my opinion it would unreasonable (and possibly lazy on your part) just to ask for an overview of the entire area. I think it would be much, much better to narrow the focus, but you will have to decide for yourself what you want to focus on and how to make a good MO question out of it. This would require some thought and possibly research on your part, which can't possibly be a bad thing.

    I find asking for "what ideas are possible to push further" an odd request. If people know which of their ideas are possible to push further, then it is overwhelmingly likely that they would go right ahead and push them further already. It would be much, much better if there were some ideas you had yourself, and wondered if they could be pushed further or had potential.

    • CommentAuthorquid
    • CommentTimeJan 28th 2013

    Possibly the following is some 'abstract' duplicate:


    If it is not, please, elaborate on what would be the difference.

    • CommentAuthorSirrush
    • CommentTimeJan 28th 2013 edited
    I get what you mean. It's not that I'm trying to be lazy but I don't know how to refine my question... I could only do that if I already knew a rough answer to it. thanks for your inputs both. That link has a lot of good info too. I will do some more research on this.

    Dear Sirrush -- I trust you when you say you're not trying to be lazy, but with regard to "I could only do that if I already knew a rough answer to it" -- welcome to the club. That's just how it is in research: we get some ideas and then try to write them out and sooner or later find we get stuck at some point, some blockage. That's when the time might be ripe to use MO! Because by that time you have become acquainted with some relevant issues, and can pinpoint a difficulty with mathematical precision. And someone who has been down that road might be able to help out, and glad to do so.

    The standards are pretty high, but the rewards are great!

    • CommentAuthorWill Jagy
    • CommentTimeJan 28th 2013
    I think you would be better off asking things on MSE. Take a look through for anything interesting
    I second what Will says.