Can the natural numbers be defined in terms of the non-trivial zeta zeros?

Question

Can the natural numbers be defined in terms of the non-trivial zeta zeros? Presumably they can, since $\pi(x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^\rho),$ and $\zeta(s)=\sum n^{-s}=\prod(1-p^{-s})^{-1} ,(s>1),$ but I have not come across it.

Answer 1

A partial answer.

The trivial zeroes of the Riemann zeta function are $\{-2,-4,-6,...\}$. So the natural even numbers are the opposite of the trivial zeroes.

Edit: The OP changed trivial to non trivial.