  We establish the large deviations principle (LDP) and the moderate deviations
principle (MDP) and an almost sure version of the central limit theorem (CLT)
for the stochastic 3D viscous primitive equations driven by a multiplicative
white noise allowing dependence on spatial gradient of solutions with initial
data in $H^2$. The LDP is established using the weak convergence approach of
Budjihara and Dupuis and uniform version of the stochastic Gronwall lemma. The
result corrects a minor technical issue in Z. Dong, J. Zhai, and R. Zhang:
Large deviations principles for 3D stochastic primitive equations, J.
Differential Equations, 263(5):3110-3146, 2017, and establishes the result for
a more general noise. The MDP is established using a similar argument.
