Description of longitudinal modes in moderately coupled Yukawa systems with the static local field correction

In moderately coupled Yukawa fluids, longitudinal mode dispersion is determined by the competition between kinetic and potential effects. In a recent paper [Khrapak and Cou\"edel, Phys. Rev. E 102, 033207 (2020)], a semi-phenomenological dispersion relation was constructed by the ad-hoc addition of the Bohm-Gross kinetic term to the generalized instantaneous excess bulk modulus, which showed very good agreement with simulations. In this paper, a nearly identical dispersion relation is derived in a rigorous manner based on a dielectric formulation with static local field corrections. At moderate coupling, this formalism is revealed to be more accurate than other successful theoretical approaches.