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I'm almost finishing my PhD in applied mathematics, but I'm planning soon (after doing post-doc) to start seriously doing research on problems about proving irrationality of real numbers. Whenever I have a chance I train myself reading proofs of this type and collecting articles and bibliography I find on internet. In 2010 I read a beautiful proof about $e$ being irrational, this is the link:

           http://math.stanford.edu/~rhoades/FILES/irrationalityE4d.pdf  

The beauty of the proof proposed in the above link is that is applicable to many other real numbers expressed in terms of infinite series.

I read the proof that $pi$ is irrational in Michael Spivak's calculus book. The proof uses elementary mathematics. I found it hard to see the intuition behind it (maybe reading Lambert's proof will show the intuition), but I was able to follow all the details and understand the contradiction exposed in the book.

Surprisingly I found an article where differential equations are used to prove irrationality of certain real numbers, this is the link:

           http://rinconmatematico.com/irracionalpi/irracpi.pdf   

I would like somebody to guide me with some literature I must follow so that I can train myself in tackling this kind of problems. What kind of books are "a must read"? "Articles"? Books that give historical context of problems and the ideas of the proofs, the intuition behind them, books written with the "heart" showing the beauty of this subject.

Also, what area of mathematics specializes in solving problems of this kind? Is it analytic number theory? Transcendental number theory?

There are so many attractive open problems like $\pi^e$, $\pi + e$ and Euler's gamma constant (my favorite!), where irrationality is not known. I found an article written by Jonathan Sondow titled "Criteria for irrationality of Euler's constant", but I couldn't follow the details. That's why I need the training.

I consider myself "not too bad" in real and complex analysis. I was able to follow proofs like the prime number theorem (In Stein's book, Complex Analysis), which requires analytic continuation of the zeta function. I know this proof by heart. Also proofs like (I found this on "An Introduction to Number Theory", Graham Everest, I like his way of writting):

                $\sum_{p \leq N} \geq \log\log N - 1$

where $p$ is prime. I like to read about finding closed formulas too, formulas of complicated series (Zeta function evaluated at even numbers), complicated definite integrals (leading to Euler's gamma constant for example)... that kind of good stuff! (I know this is connected with the irrationality proofs, based on the first link I gave, I'm sure!).

Thanks for the help!

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One article I really like is: A Geometric Proof that e Is Irrational and a New Measure of Its Irrationality. Jonathan Sondow. The American Mathematical Monthly, Vol. 113, No. 7 (Aug. - Sep., 2006), pp. 637-641. jstor.org/stable/27642006. I don't see an easy way of generalizing this geometric method for fundamentally different irrational real numbers, but the paper is a fun read (and you mentioned Sondow in your question). A piece of unsolicited advice, though: those "attractive open problems" are open for a reason; don't get stuck on something intractable. –  Benjamin Dickman Oct 25 '12 at 4:31
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I think you should definitely look at Baker's book, "Transcendental Number Theory" books.google.ca/books?id=SmsCqiQMvvgC Beyond that, I suspect you really ought to get a mentor who is a specialist in this area and knows where some doable problems might be found. mathoverflow.net is probably not a feasible substitute for such a mentor. –  Robert Israel Oct 25 '12 at 7:00
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I'd have to second Robert Israel's advice about finding a mentor; there is much about this post that sounds like a youthful dream or Jugendtraum (an impression strengthened by phrases like "I know this proof by heart"). The actual problems mentioned are apparently really, really hard, and changing a career trajectory without having a real inside edge on such problems sounds somewhat risky to me. I am reminded of a phrase in English warning about "putting all of one's eggs in one basket". –  Todd Trimble Oct 25 '12 at 13:20
    
+100 @Todd Trimble –  Yemon Choi Oct 27 '12 at 11:37
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5 Answers 5

This is quite vague and broad a question, but one suggestion.

Have a look around at Michel Waldschmidt's webpage (if you can read French it will be even still better) containing among others lot of expository work, which in part is accessible.

For example a very recent presentation (the file is almost 5Mb, as there are many pictures) Transcendental Number Theory: recent results and open problems

or also Transcendence of Periods (also about 5Mb).

A rather recent (2000) survey

Un demi-siècle de trancedence in particular look at the references there.

And a lot more, and also look under 'enseignement' (ie, teaching).

Reagrding naming: another important keyword is Diophantine Approximation and there is also Metric Number Theory sort of in that direction.

However, names are names, and what actual mathematics you will need precisely could vary a lot. Some things but not all are a lot more 'algebraic-geomtric' than 'analytic', very roughly speking.

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You were talking about proofs of irrationality. However pi and e are known to be transcendental, and the proof of transcendence is usually harder than a proof of irrationality. Differential equations is indeed a useful tool in proving transcendence. Look at any book which has transcendental numbers in its title.

The key authors are Baker, Mahler, Shidlovski, Gelfond and Siegel.

However, if we are talking about the proofs of IRRATIONALITY (that is of numbers which were not known to be irrational), here is a famous and relatively simple proof:

Apéry, Roger Irrationalité de $\zeta(2)$ et $\zeta(3)$. Astérisque 61, 11-13 (1979).

Then you can look at the papers which cite this great result.

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Well, I beg to strongly disagree with the statement that differential equations are "an indispensable tool in proving transcendence". –  Salvo Tringali Oct 22 '13 at 19:12
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To add to quid's suggestion (look around Michel Waldschmidt's webpage), I'd say specifically that Waldschmidt's slides on Schanuel's conjecture are probably worth looking at. This may be more for one's general education about transcendental number theory than any suggestion for research, since Schanuel's conjecture is widely regarded as a Holy Grail of transcendental number theory, and a proof seems to be nowhere in sight. It is, however, quite beautiful!

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You might be interested in the concept called periods. Quid already mentioned the work of Waldschmidt, but see also Wikipedia, and in particular the link at the bottom to the article of Kontsevich and Zagier.

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I'd like to add the book for one of the two number courses I've taken which was an entire senior seminar (last year of undergraduate) on transcendental numbers. Making Transcendence Transparent by Burger and Tubbs was good read and stressed the basic structure of the proofs by using sidebars to explain the ideas next to the formal proof. To this day the book contains my favorite theorem: $\pi \neq \frac{22}{7}$. Because this book is aimed at American undergraduates it is more conversational than a good for graduate students or beyond.

As a bonus, it appears to be less expensive than I remembered.

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