[Math] Why is Catalan’s constant $G$ important

Question

I am aware that Catalan's constant appears in the evaluation of many definite integrals, as well as in the evaluation of certain infinite series, and is a special value of a function closely related to the Riemann zeta function, and so on.

But is there a way in which we might think of this constant as important in its own right, and not come at it 'indirectly;' not approaching it only through other 'loftier' concepts first, or using it as a convenient shorthand for the numerical result of a certain integration or summation process, etc., but through some simple concepts that first lead you to the idea of some important constant, whose more sophisticated properties we can then deduce, and in doing so derive its relations to the transcendental functions, integral and series representations, so that we arrive at the idea of Catalan's constant itself as some important concept in its own right, without appealing to it as kind of a secondary idea?

For example, we can do this with $\pi$ and $e$ quite easily. The concept of $\pi$ appears as a simple geometric idea. I only need to understand some basic geometrical ideas, and then it would be only a matter of time before I arrived at the idea of $\pi$ and understood its importance, even if I didn't fully understand the full depth of its importance. The same with $e,$ or even the Euler-Mascheroni constant $\gamma.$ I only need some basic concepts of calculus.

With more sophisticated concepts I obviously need more sophisticated prerequisites, but regardless, it is often the case that many fundamental ideas actually have a very simple, and very intuitive germ, which is why you see them pop up again and again, and often there is still some fairly simple relationship between ideas which tells you immediately that something may be important, even if those ideas are themselves not quite 'basic'.

With Catalan's constant, I haven't seen any kind of explanation that explains this. It's almost always 'Catalan's constant appears in the evaluation of …' which only tells you that it is important, but not really where this importance comes from, or why you would be led to care about this constant as anything more than a useful notation for some integral or sum, unlike many other ubiquitous objects in mathematics, where you might first grasp them as an important idea, and then notice their many other uses.

Can anyone explain why Catalan's constant is important, or why you would be motivated to care about this constant in its own right? I appreciate that this question may be vague but I'd love to hear any good answers.

Answer 1

For myself I would like to bring up Leonard Lewin's Polylogarithms and associated function. In chapter $2$, dealing with the Inverse Tangent Integral, Catalan's Constant is introduced:

The value of $\operatorname{Ti}_2(1)$ cannot, however, be deduced from this functional relation, and so far as is known is a new constant of analysis, denoted by $G$ known as Catalan's constant. It occurs in various contexts, and its value, to $8$ decimal places, is

$$\operatorname{Ti}_2(1)~=~G~=~0.91596559\tag{2.7}$$

Fascinating about this constant I would claim is especially its broad occurrence within many, on the first sight distinct, problems of closed-form integration aswell as closed-form summation. As José Carlos Santos already pointed out: whilst taking into account how less we know about this real number it is odd how widely it can be used.

I firstly encoutered this constant in the context of Dilogarithms, to be precise by invoking auxiliary functions such as the aforementioned Inverse Integral Tangent or the Clausen Functions, and the Dirichlet Beta Function which both tend to be possible defintions for the constant in terms of an infinite series, namely

$$G~=~\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^2}\tag1$$

Even though this sum looks pretty similiar to the Riemann Zeta Function and its relatives it cannot be related to them that simple $($the only possible functional relation I am aware of is given within this article but hence it uses the Polygamma Functions aswell I would not call this relation "easy" as e.g. the one between the Riemann Zeta Function and the Dirichlet Eta Function$)$. To speak for myself the different "character" of the Dirichlet Beta Function, i.e. having expressible values for odd positive integers, not being expressable with the help of the Riemann Zeta Function alone, etc., justifies its importance.

Of course, only to appear everywhere is not a criterion for being important alone but appearing over and over again, especially in connection with $\pi$, seems to imply that there is something more about this number. Concerning the appearance I would like to refer to this question here on MSE dealing with the relationship between Catalan's Constant and $\pi$ in particular.

To bring up another point showing the importance of Catalan's Constant I would claim that its role can be compared with the role of Apéry's Constant $\zeta(3)$. Both can be defined in terms of infinite series, namely the Dirichlet Beta Function and the Riemann Zeta Function respectively. Both are the first positive integers for which the underlying Function cannot be expressed using other constants. As already mentioned there are formulae for the even positive integer values of the Riemann Zeta Function and for the odd positive integer values of the Dirichlet Beta Function given by

\begin{align*} \zeta(2n)~&=~(-1)^{n-1}\frac{(2\pi)^n}{2(2n)!}\operatorname{B}_{2n}\tag2\\ \beta(2n+1)~&=~(-1)^n\frac{\pi^{2n+1}}{4^{n+1}(2n)!}\operatorname{E}_{2n}\tag3 \end{align*}

Whereas we defined $\beta(2)$ as Catalan's Constant and $\zeta(3)$ as Apéry's Constant. A crucial different, however, is that we know that the latter constant is in fact irrational due to Apéry who proved this back in $1979$. Interesting enough a similiar series representation for $G$ exists like the one Apéry used within his proof. These series are given by

\begin{align*} \zeta(3)~&=~\frac52\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^3\binom{2n}n}\tag4\\ \beta(2)~&=~\frac\pi8\ln(2+\sqrt 3)+\frac38\sum_{n=0}^\infty \frac1{(2n+1)^2\binom{2n}n}\tag5 \end{align*}

All in all Catalan's Constant does not only occurs in a bunch of mathematical problems related to integration and summation but also plays a quite important role in the field of Zeta Functions and relatives. I would say especially the striking parallels between Apéry's Constant and Catalan's Constant consolidate the importance of this constant.