  We investigate the properties of the recently observed "demon" mode, a 3D
acoustic plasmon, in Sr$_2$RuO$_4$ with an emphasis on evaluating its role for
the pairing interactions in this superconductor. The demon mode is a low-energy
electronic excitation, and it has been suggested that it could contribute to a
reduced Coulomb repulsion and even a possible attractive interaction between
electrons. In this study, we explicitly calculate the dynamically screened
Coulomb interaction for Sr$_2$RuO$_4$ by using a renormalized tight-binding
band structure and the random phase approximation for the dielectric function.
Although the focus here is on Sr$_2$RuO$_4$, this material is considered mainly
as a prototype system, having an observed demon mode, and our results should be
considered as a guide for application to other systems. Our calculations show
that there are regions in ($\mathbf{q}$, $\omega$) space where the Coulomb
interaction becomes attractive. We find that, although the demon mode is not
capable of producing a total attractive electron pairing interaction in
Sr$_2$RuO$_4$, it does contribute to a significant reduction in the Coulomb
repulsion at the relevant pairing energy scale.
