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.