Why is it hard to prove that the Euler Mascheroni constant is irrational?

Question

Philosophically why should proving that $\gamma$ is irrational (let alone transcendental) be so much harder than proving $\pi$ or $e$ are irrational?

Answer 1

There are number theorists who understand this subject much better than I do. However, I feel obliged to post an incomplete answer quickly before people have a chance to close this question.

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There are a lot more connections known between $\pi$ and $e$ and other numbers than between $\gamma$ and other numbers. We can get proofs of their irrationality by using some of these connections, such as continued fraction expansions for both.

$\gamma$ may be thought of as a renormalized version of $\zeta(1)$, where $\zeta$ is the Riemann zeta function $\zeta(s) = \sum_{n=1}^\infty n^{-s}$.

$$\gamma = \lim_{s\to 1} \bigg(\zeta(s) - \frac{1}{s-1} \bigg)$$

At even integers, $\zeta(s)$ may be rewritten as a sum over nonzero integers, not just the positive integers. That's one explanation for why it is easier to get a handle on $\zeta(s)$ at even values (where it is a rational times $\pi^s$) than at positive odd integer values. See the answers to "Establishing zeta(3) as a definite integral and its computation."

There is some hope. Apéry proved that $\zeta(3)$ is irrational, and this can be related to proofs that other well known numbers are irrational. There are expressions for $\pi$, $\log 2$, $\zeta(3)$ as periods, definite integrals of algebraic functions on $[0,1]$. These can be used in a unified way to prove all of these are irrational (although it's still tricky for $\zeta(3)$), and there are conjectures about the possible rational or algebraic relations between periods. However, so far, $\gamma$ isn't known to be a period although it is an exponential period (as is $e$). No other values of $\zeta$ at positive odd integers are individually known to be irrational.

Answer 2

Philosophically, there is essentially only one way to prove that a number is irrational/transcendental, which is to use the fact that there is no integer between 0 and 1. That is, one assumes that the number in question is rational/algebraic, and constructs some quantity that can be shown to be bounded away from 0, less than 1, and also an integer. To get these estimates, one typically needs some rapidly converging series expansion that is closely related the number of interest. For example the reason that Fourier's proof that $e$ is irrational is so simple is that we have a ready-made rapidly converging series $\sum 1/n!$ for $e$ that allows us to construct an integer between 0 and 1 from the assumption that $e$ is rational. As for numbers like $\pi$ or $e^\pi$, they basically piggyback on the rapidly converging series $\exp z = \sum z^n/n!$ because $\exp(i\pi) = -1$.

In general, the less obvious it is how to relate your number to a suitable rapidly converging series, the harder it will be to prove irrationality/transcendence. Apéry's dramatic success with $\zeta(3)$ was based on the highly non-obvious rapidly converging series representation $$ \zeta(3) = \frac{5}{2} \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^3\binom{2n}{n}}. $$ But $\zeta(5)$ doesn't yield to exactly the same method because there is ample numerical evidence that the obvious analogous series that you would conjecture for it after seeing the above identity does not exist. Later progress on $\zeta(2n+1)$, particularly by Rivoal and Zudilin, relies on taking various subtle linear combinations of them in order to eventually deduce the nonexistent integer between 0 and 1, and so are only able to prove that at least one of a set of zeta values is irrational.

So could there be another proof that Euler missed out there, based on an elementary but non-obvious identity for $\gamma$ that has the right properties for the usual machinery to grind through? Maybe, but currently there is no method in sight for relating $\gamma$ in the right way to a suitable rapidly converging series. If you like, that is the "philosophical" reason why we're stuck.

By the way, I'd highly recommend Making Transcendence Transparent by Burger and Tubbs if you want an accessible treatment of transcendental number theory. They do an excellent job of showing how the same basic ideas underlie all the results in the area, while introducing the technical complications one at a time in digestible chunks.