  We show that the three-dimensional map between fermions and bosons at finite
temperature generalises for all odd dimensions $d>3$. We further argue that
such a map has a nontrivial large $d$ limit. Evidence comes from studying the
gap equations, the free energies and the partition functions of the $U(N)$
Gross-Neveu and CP$^{N-1}$ models for odd $d\geq 3$ in the presence of
imaginary chemical potential. We find that the gap equations and the free
energies can be written in terms of the Bloch-Wigner-Ramakrishnan $D_d(z)$
functions analysed by Zagier. Since $D_2(z)$ gives the volume of ideal
tetrahedra in 3$d$ hyperbolic space our three-dimensional results are related
to resent studies of complex Chern-Simons theories, while for $d>3$ they yield
corresponding higher dimensional generalizations. As a spinoff, we observe that
particular complex saddles of the partition functions correspond to the zeros
and the extrema of the Clausen functions $Cl_d(\theta)$ with odd and even index
$d$ respectively. These saddles lie on the unit circle at positions remarkably
well approximated by a sequence of rational multiples of $\pi$.
