  Electrical conductivity is an important transport response in
superconductors, enabling clear signatures of dynamical interactions to be
observed. Of primary interest in this paper is to study characteristics of the
electron-phonon interaction in weak-coupling Eliashberg theory (Eth), and to
note the distinctions with Bardeen-Cooper-Schrieffer (BCS) theory. Recent
analysis of weak-coupling Eth has shown that while there are modifications from
the BCS results, certain dimensionless ratios are in agreement. Here we show
that the conductivities in BCS theory and Eth fundamentally differ, with the
latter having an imaginary gap component that damps a divergence. We focus on
the dirty limit, and for both BCS theory and Eth we derive expressions for the
low-frequency limit of the real conductivity. For Eth specifically, there are
two limits to consider, depending on the relative size of the frequency and the
imaginary part of the gap. In the case of identically zero frequency, we derive
an analytical expression for the nuclear magnetic resonance relaxation rate.
Our analysis of the conductivity complements the previous study of the Meissner
response and provides a thorough understanding of weak-coupling Eth.
