A (3+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
is considered, which can be used to describe many nonlinear phenomena in plasma
physics. By virtue of binary Bell polynomials, a bilinear representation of the equation
is succinctly presented. Based on its bilinear formalism, we construct soliton solutions
and Riemann theta function periodic wave solutions. The relationships between
the soliton solutions and the periodic wave solutions are strictly established and the
asymptotic behaviors of the Riemann theta function periodic wave solutions are analyzed
with a detailed proof.