  We examine an Unruh-DeWitt particle detector coupled to a scalar field in
three-dimensional curved spacetime. We first obtain a regulator-free expression
for the transition probability in an arbitrary Hadamard state, working within
first-order perturbation theory and assuming smooth switching, and we show that
both the transition probability and the instantaneous transition rate remain
well defined in the sharp switching limit. We then analyse a detector coupled
to a massless conformally coupled field in the Hartle-Hawking vacua on the
Banados-Teitelboim-Zanelli black hole, under both transparent and reflective
boundary conditions at the infinity. A selection of stationary and
freely-falling detector trajectories are examined, including the co-rotating
trajectories, for which the response is shown to be thermal. Analytic results
in a number of asymptotic regimes, including those of large and small mass, are
complemented by numerical results in the interpolating regimes. The boundary
condition at infinity is seen to have a significant effect on the transition
rate.
