  We study the Shannon mutual information in one-dimensional critical spin
chains, following a recent conjecture (Phys. Rev. Lett. 111, 017201 (2013)), as
well as R\'enyi generalizations of it. We combine conformal field theory
arguments with numerical computations in lattice discretizations with central
charge $c=1$ and $c=1/2$. For a periodic system of length $L$ cut into two
parts of length $\ell$ and $L-\ell$, all our results agree with the general
shape-dependence $I_n(\ell,L)=(b_n/4)\ln \left(\frac{L}{\pi}\sin \frac{\pi
\ell}{L}\right)$, where $b_n$ is a universal coefficient. For the free boson
CFT we show from general arguments that $b_n=c=1$. At $c=1/2$ we conjecture a
result for $n>1$. We perform extensive numerical computations in Ising chains
to confirm this, and also find $b_1\simeq 0.4801629(2)$, a nontrivial number
which we do not understand analytically. Open chains at $c=1/2$ and $n=1$ are
even more intriguing, with a shape-dependent logarithmic divergence of the
Shannon mutual information.
