A complex number $z$ is an integer if and only if $\sin(\pi z)=0$.


It follows that a complex number $z$ is an integer if and only $\sin^2(\pi z) = 0$. So for a **real** analytic function $f$ and any **real** number $x$, $x$ and $f(x)$ are both integers if and only if $\sin^2(\pi x) + \sin^2(\pi f(x)) = 0$.


Are there analytic functions that do the same over the complex numbers? Given a (complex) analytic function $f$, is there a way to assemble an analytic function $F$ so that $z$ and $f(z)$ are both integers if and only if $F(z) = 0$?



Analytical predicate for integers over complex numbers
[Reposted as an answer, as requested. -T.]


While this can be done for any given $f$ (as indicated by Angelo's answer), it can't be done in a way which is stable with respect to perturbations of $f$ (which rules out most "analytic" answers, such as the one in the real case). This is because zeroes of a complex analytic function are stable wrt perturbations (Rouche's theorem), whereas the set of $z$ for which $z$ and $f(z)$ are both integers is unstable (generically this set is empty). (In the real case, of course, zeroes are unstable wrt perturbation, especially for functions that are sums of squares.)


