  In this work, we investigate the dynamics of the number density fluctuations
of a dilute suspension of active particles in a linear viscoelastic fluid. We
propose a model for the frequency-dependent diffusion coefficient of the active
particles, which captures the effect of rotational diffusion on the persistence
of their self-propelled motion and the viscoelasticity of the medium. Using
fluctuating hydrodynamics, the linearized equations for the active suspension
are derived, from which we calculate its dynamic structure factor and the
corresponding intermediate scattering function. For a Maxwell-type rheological
model, we find an intricate dependence of these functions on the parameters
that characterize the viscoelasticity of the solvent and the activity of the
particles, which can significantly deviate from those of an inert suspension of
passive particles and of an active suspension in a Newtonian solvent. In
particular, in some regions of the parameter space we uncover the emergence of
oscillations in the intermediate scattering function at certain wave numbers,
which represent the hallmark of the non-equilibrium particle activity in the
dynamical structure of the suspension and also encode the viscoelastic
properties of the medium.
