I'm almost finishing my PhD in applied mathematics, but I'm planning soon (after doing a 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: Irrational numbers arising from certain differential equations by M. Ram Murty and V. Kumar Murty.

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 on 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" by Graham Everest; I like his way of writing): $$\sum_{p \leq N} \frac{1}{p} \geq \log\log N - 1$$ where $p$ is a prime-valued variable. 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!).