Dynamical polarization in ABC-stacked multilayer graphene in a magnetic field

In the continuum low energy model, we calculate the one-loop dynamical polarization functions in ABC-stacked (rhombohedral) $n$-layer graphene in a magnetic field. Neglecting the trigonal warping effects, they are derived as functions of wavevector and frequency at finite chemical potential, temperature, band gap, and the width of Landau levels. The analytic results are given in terms of digamma functions and generalized Laguerre polynomials and have the form of double sums over Landau levels. Various particular limits for polarization functions (static, clean, etc.) are discussed. The intralayer and interlayer screened Coulomb potentials are numerically calculated as functions of momentum and frequency.