Expressing functions using the arithmetic dictionary
I'm not sure this is exactly what you are looking for, but you might find what you want in Theorem 3.2.1 of Computability Theory by S. Barry Cooper:


  All recursive functions are representable in PA.


that is for any recursive function $f$, there exists a binary predicate $F$ in the language of arithmetic such that for any natural numbers $x$ and $y$ we have
$$ f(x) = y ~\Rightarrow~ \vdash_{PA} F(x,y) $$
and
$$ f(x) \neq y ~\Rightarrow~ \vdash_{PA} \lnot F(x,y) $$
where $\vdash_{PA}$ means 'PA proves'.

This theorem is central to Gödel's famous incompleteness theorem, so you might also want to take a look at ch. 8 of the mentioned book where it is discussed, and this notion of 'representability' is extended to 'semi-representability', to include the c.e. sets as well.