It's well known that the computation models untyped lambda calculus and $\mu$-recursive function are equivalent in terms of computability (in fact they are both Turing complete.) It is also well known that, among some of the variations of them, the simply typed lambda calculus ($\lambda_{\rightarrow}$) and the primitive recursive function ($R_p$), are less powerful (i.e. neither of them is Turing complete.)
Since the intuition of the essential restrictions applied in both cases feels similar, I have the instinct that the class of functions computable by either of them is the same, i.e. $\lambda_{\rightarrow} \cong R_p $; but I cannot find any literature discussing this issue myself after a quick search.
Thus, I'd like to ask 1) whether simply typed lambda calculus and primitive recursive function are equivalent and 2) if so, is there any literature that I can have a look at discussing this?