The total recursive functions are exactly those number-theoretic functions that can be represented by a $\Sigma_1$ formula of first-order arithmetic.

Is there a similar characterization of the *primitive* recursive functions? I'm looking for something like for example

(wild conjecture)The primitive recursive functions are those that can be represented by a $\Sigma^0_1$ formula which can be proved total and single-valued by $\Delta^0_0$ induction.