Recursive functions and existentially closed structures

The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory $T$ in which all partially recursive functions are representable, yet $T$ does not interpret Robinson's theory $R$. To this end, we borrow tools from model theory---specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of $\exists\forall$ theories interpretable in existential theories in the process.