## Catalan’s Constant and periods

There is a 60th birthday conference in honour of Frits Beukers in Utrech in July; I’m hoping to swing by there on the way to Oberwolfach. Thinking about matters Beukers made me reconsider an question that I’ve had for while.

There is a fairly well known explanation of why $\zeta(3)$ should be irrational (and linearly independent of $\pi^2$) in terms of Motives. There is also a fairly good proof that $\zeta(3) \ne 0$ in terms of the non-vanishinjg of Borel’s regulator map on $K_5(\mathbf{Z})$. (I guess there are also more elementary proofs of this fact.) A problem I would love to solve, however, is to show that, for all primes $p$, the Kubota-Leopoldt $p$-adic zeta function $\zeta_p(3)$ is non-zero. Indeed, this is equivalent to the injectivity of Soule’s regulator map

$K_5(\mathbf{Z}) \otimes \mathbf{Z}_p \rightarrow K_5(\mathbf{Z}_p).$

(Both these groups have rank one, and the cokernel is (at least for $p > 5$) equal to $\mathbf{Z}_p/\zeta_p(3) \mathbf{Z}_p$ by the main conjecture of Iwasawa theory.) It is somewhat of a scandal that we can’t prove that $\zeta_p(3)$ is zero or not; it rather makes a mockery out of the idea that the “main conjecture” allows us to “compute” eigenspaces of class groups, since one can’t even determine if there exists an unramified non-split extension

$0 \rightarrow \mathbf{Q}_p(3) \rightarrow V \rightarrow \mathbf{Q}_p \rightarrow 0$

or not. Well, this post is about something related to this but a little different. Namely, it is about the vaguely formed following question:

What is the relationship between a real period and its $p$-adic analogue?

Since one number is (presumably) in $\mathbf{R} \setminus \mathbf{Q}$ and the other in $\mathbf{Q}_p \setminus \mathbf{Q}$, it’s not entirely clear what is meant by this. So let me give an example of what I would like to understand. One could probably do this example with $\zeta(3)$, but I would prefer to consider the “simpler” example of Catalan’s constant. Here

$G = \displaystyle{\frac{1}{1} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} \ldots } = L(\chi_4,2) \in \mathbf{R},$

is the real Catalan’s constant, and

$G_2 = L_2(\chi_4,2) \in \mathbf{Q}_2$

is the $2$-adic analogue. (There actual definition of the Kubota-Leopoldt zeta function involves an unnatural twist so that one could conceivably say that $L_2(\chi_4,2) = 0$ and that the non-zero number is $\zeta_2(2)$, but this is morally wrong, as the examples below will hopefully demonstrate. Morally, of course, they both relate to the motive $\mathbf{Q}(2)(\chi_4)$.)

So what do I mean is the “relation” between $G$ and $G_2$. Let me give two relations. The first is as follows. Consider the recurrence relation (think Apéry/Beukers):

$n^2 u_n = (4 - 32 (n-1)^2) u_{n-1} - 256 (n-2)^2 u_{n-2}.$

It has two linearly independent solutions with $a_1 = 1$ and $a_2 = -3$, and $b_1 = -2$ and $b_2 = 14$. One fact concerning these solutions is that $b_n \in \mathbf{Z}$, and $a_n \cdot \mathrm{gcd}(1,2,3,\ldots,n)^2 \in \mathbf{Z}.$ Moreover one has that:

$\displaystyle{ \lim_{n \rightarrow \infty} \frac{a_n}{b_n}} = G_2 \in \mathbf{Q}_2.$

The convergence is very fast, indeed fast enough to show that $G_2 \notin \mathbf{Q}$. What about convergence in $\mathbf{R}$, does it converge to the real Catalan constant? Well, a numerical test is not very promising; for example, when $n = 40000$ one gets $0.625269 \ldots$, which isn’t anything like $G = 0.915966 \ldots$; for contrast, for this value of $n$ one has $a_n/b_n - G_2 = O(2^{319965})$, which is pretty small. There are, however, two linearly independent solutions over $\mathbf{R}$ given analytically by

$\displaystyle{\frac{(-16)^n}{n^{3/2}} \left( 1 + \frac{5}{256} \frac{1}{n^2} - \frac{903}{262144} \frac{1}{n^4} + \frac{136565}{67108864} \frac{1}{n^6} - \frac{665221271}{274877906944} \frac{1}{n^8} + \ldots \right)},$

\begin{aligned} \frac{(-16)^n \cdot \log n}{n^{3/2}} \left( 1 + \frac{5}{256} \frac{1}{n^2} - \frac{32261}{7864320} \frac{1}{n^4} + \frac{136565}{67108864} \frac{1}{n^6} - \frac{665221271}{274877906944} \frac{1}{n^8} + \ldots \right)\\ +\frac{(-16)^n}{n^{3/2}} \left( -\frac{1}{768} \frac{1}{n^2} + \frac{32261}{7864320} \frac{1}{n^4} - \frac{30056525}{8455716864} \frac{1}{n^6} + \frac{1778169492137}{346346162749440} \frac{1}{n^8} + \ldots \right) \end{aligned},

from which one can see that $a_n/b_n$ must converge very slowly, and indeed, one has (caveat: I have some idea on how to prove this but I’m not sure if it works or not):

$\displaystyle{\frac{a_n}{b_n} = G - \frac{1}{(0.2580122754655 \ldots) \cdot \log n + 0.7059470639 \ldots}}$

So one has a naturally occurring sequence which converges to $G$ in $\mathbf{R}$ and $G_2$ in $\mathbf{Q}_2$. So that is some sort of “relationship” alluded to in the original question. Here’s another connection. Wadim Zudilin pointed out to me the following equality of Ramanujan:

$\displaystyle{G = \frac{1}{2} \sum_{k=0}^{\infty} \frac{4^k}{(2k + 1)^2 \displaystyle{\binom{2k}{k}}}} \in \mathbf{R}.$

This sum also converges $2$-adically. So, one can naturally ask whether

$\displaystyle{G_2 =^{?} \frac{1}{2} \sum_{k=0}^{\infty} \frac{4^k}{(2k + 1)^2 \displaystyle{\binom{2k}{k}}}} \in \mathbf{Q}_2.$

(It seems to be so to very high precision.) These are not random sums at all. Indeed, they are equal to

$\displaystyle{ \frac{1}{2} \cdot F \left( \begin{array}{c} 1,1,1/2 \\ 3/2,3/2 \end{array} ; z \right)}$

at $z = 1$. Presumably, both of these connections between $G$ and $G_2$ must be the same, and must be related to the Picard-Fuchs equation/Gauss-Manin connection for $X_0(4)$. This reminds me of another result of Beukers in which one compares values of hypergeometric functions related to Gauss-Manin connections and elliptic curves, and finds that they converge in $\mathbf{R}$ and $\mathbf{Q}_p$ for various $p$ to algebraic (although sometimes different!) values. Of course, things are a little different here, since the values are (presumably) both transcendental. Yet it would be nice to understand this better, and see to what extent there is a geometric interpretation of (say) the non-vanishing of $L_p(\chi,2)$ for some odd quadratic character $\chi$. Of course, one always has to be careful not to accidentally prove Leopoldt’s conjecture in these circumstances.