The demise of the arxiv continues into 2008 with yet another (cough)
disproof of the Riemann Hypothesis (reported by
Lubos). Elementary disproofs seem popular these days. Since
Connes tells us the Riemann Hypothesis is closely related to Quantum Gravity, that means Quantum Gravity must be Elementary also. Elementary in the sense of axiomatically foundational, maybe?
Yesterday we came across categorified cardinalities once again. For example, to compute the cardinality of the groupoid of finite sets we just need to sum the cardinalities of the groupoid components,

FinSet0 = $\frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots = e$
The Riemann zeta function, for real arguments, looks a bit like such a sum, namely
$\zeta (s) = \frac{1}{1^{s}} + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \cdots$
so when $s$ is a positive integer this might measure the cardinality of the sequence of products of cyclic groups $( \mathbb{Z}_{n} )^{s}$ for $n \in \mathbb{N}$. What sort of groupoid is this? It is very reminiscent of
Rota's ideas on profinite combinatorics and the Riemann zeta function. Hmmm. We
know that $\zeta (2) = \frac{\pi^{2}}{6}$ and so on, so the factors of $\pi$ must come from a cardinality for such a groupoid. The question is, what basic thing has (products of) cyclic automorphism groups? One possibility is oriented
polygons and we already know that $n$gons are associated with $n3$ dimensional associahedra, and associahedra are related to the permutohedra, the vertices of which give the elements of the groups counted by $e$.
This seems like such a nice way to relate $e$ and $\pi$ and $1 = e^{i \pi}$.