How hard is the proof of $\pi$ or $e$ being transcendental?

I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious what types of techniques are used and just how difficult of a problem it is. Would this result be a reasonable capstone to any course? (any course that isn’t essentially “how to prove…”)

Another part of this is the following observation: as time passes deep results become easier to understand or rather assimilate into ones body of knowledge and some problems are just hard. I am wondering if people feel like this result is something that a grad student could spend some leisure time and understand, or if it really is something only graspable by “experts” (meaning people in the appropriate field and not a general mathematical audience). How specialized are the techniques used for the problem at hand? Have they been used to prove different results? are the techniques drastically different for $e$ and $\pi$?

Hope this isn’t too soft of a question. I was talking with my roommate, also a math grad student, and it came up. I said that it was a classically difficult result, but then wondered if that was right, so here I am.

Note: I don’t want a proof or a sketch of one, but maybe a heuristic as to why new techniques were needed or explaining the troubles one has when being naive or using early methods to attack the problem.

Edit: As Matt E pointed out I should also ask: what are the old techniques?
Also, did I even tag the question correctly? These seem to be the areas I would put them in, but I don’t know the anything about this stuff.

In the above it also isn’t clear that I am wondering if the proof of this result is gotten by some clever new trick or by lots of hard had work that people couldn’t/didn’t do before? essentially, is it all elbow grease, or some clever new machinery or something completely different?

Answer

This is an answer elaborating on my comment above, and largely addressing the
edits to the original question:

I don’t think there were any “old techniques” before Hermite proved
transcendence of $e$ in the early 1870s. As far as I know, the only transcendental numbers known before then were Liouville’s interesting but
somewhat artificial examples from early that century.

The subject of transcendence, and the related subject of Diophantine approximation (i.e. approximating irrational numbers, especially irrational algebraic numbers, by rationals), is relatively new. Liouville proved the first results showing that it is not so easy to approximate an irrational algebraic number by rational ones, and used this to construct his transcendental numbers, which he could recognize as being transcendental because they are too well approximated by rational numbers. (“Too well” and “not so easy” here refer to the following problem: if you try to approximate $\alpha$ by the rational number $p/q$, can you get within a distance of $O(q^{-n})$ for some given $n$ as you let the denominator $q$ get arbitrarily large. (The larger $n$ is, the smaller $q^{-n}$ is, and so the better the rate of approximation.) Liouville showed, using the pigeon hole principal more-or-less, that if $\alpha$ is algebraic of degree $d$ then you can’t do better than $n = d$.)
But this left open the problem of showing that various given numbers (like $e$ and $\pi$) are transcendental.

If you like, here is one way to think of the problem: if you want to show that $\alpha$ is transcendental, then
you want to show that $f(\alpha) \neq 0$ for any non-zero polynomial $f$
with rational coefficients. The difficulty is that there will be lots of polynomials with real coefficients that have $\alpha$ as a root, and any one of them can be approximated as closely as you like by an $f$ with rational coefficients, so we can find (lots of!) $f$ with rational coefficients such that $f(\alpha)$ is as close to zero as we like.

So you have to find some way to pin down the difference between $f(\alpha)$ being zero and $f(\alpha)$ being very close to zero. This is not so easy!
(For example, computationally, you can’t tell the difference between $0$ and any real number that is smaller than your computational accuracy can recognize.)
And now one sees why Diophantine approximation ideas of the type mentioned above
are relevant. They are related to quantifying how close we can make $f(\alpha)$ to zero while bounding the denominators of the rational numbers involved.

It is not coincidence that bounding the denominators is relevant: morally this is an attempt to pass from working over $\mathbb Q$ to working over $\mathbb Z$.
Why do we want to do this? Well, as I already noted, it’s pretty hard to tell the difference between $\mathbb Q$ and $\mathbb R$, since the former is dense in the latter, but we can tell the difference between $\mathbb Z$ and $\mathbb R$, since the former is discrete: a non-zero integer is some definite positive distance (i.e. at least 1) away from $0$.

The preceding remarks are somewhat philosophical, and they reflect my (limited) experience of thinking about these kinds of questions. If you look at the proofs in the link above, it may not be obvious that they are relevant, but I believe that they in fact do have some relevance: e.g. you will see that the arguments reduce to considering integer rather than rational polynomials, and that growth considerations play a key role. Another thing you will see is that certain auxiliary polynomials enter the proof, and a key fact about them is that they have a high order of vanishing at their zeroes. The appearance of auxiliary polynomials, often with a high order of vanishing, is ubiquitous in this theory.

One more (somewhat cultural) remark:
Roth’s theorem, for which he got the Fields medal, is the ultimate strengthening of Liouville’s theorem: he shows that if $\alpha$ is irrational algebraic, then one can’t do better than $O(q^{-2})$ in the problem of Diophantine approximation discussed above. The proof involves (among other things) constructions with auxiliary polynomials. So my impression is that Liouville, Hermite, and Lindemann (and there are probably other names that should be here) invented a new subject, namely Diophantine approximation and transcendence theory, whose modern methods are an outgrowth of the methods that they introduced.

P.S. Reading the first few pages of Baker’s book Transcendental Number Theory
(Google Books link) might help.

Attribution
Source : Link , Question Author : Sean Tilson , Answer Author : J. M. ain’t a mathematician

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