  In this paper, we explore the low Mach number singular limit of the
local-in-time strong solutions to the compressible primitive equations with
gravity for general adiabatic coefficient. First we construct the uniform
estimate for the solutions to the non-dimensional compressible primitive
equations with general ill-prepared initial data. Due to the effects of gravity
and the anisotropy of the system, the operator with large coefficient in this
model is not explicitly skew-symmetric. Thus, obtaining the uniform estimate
requires novel techniques. After that, we investigate rigorously the low Mach
number limit of the compressible primitive equations with both well-prepared
and ill-prepared initial data. The limiting system is shown to be the
incompressible primitive equations with inhomogeneous density that depends on
the vertical variable.
