  We compute the information theoretic von Neumann entropy of the state
associated to the fermionic second quantization of a spectral triple. We show
that this entropy is given by the spectral action of the spectral triple for a
specific universal function. The main result of our paper is the surprising
relation between this function and the Riemann zeta function. It manifests
itself in particular by the values of the coefficients $c(d)$ by which it
multiplies the $d$ dimensional terms in the heat expansion of the spectral
triple. We find that $c(d)$ is the product of the Riemann xi function evaluated
at $-d$ by an elementary expression. In particular $c(4)$ is a rational
multiple of $\zeta(5)$ and $c(2)$ a rational multiple of $\zeta(3)$. The
functional equation gives a duality between the coefficients in positive
dimension, which govern the high energy expansion, and the coefficients in
negative dimension, exchanging even dimension with odd dimension.
