Interdefinability and compatibility in certain o-minimal expansions of the real field

Let us say that a real function f is o-minimal if the expansion (R,f) of the real field by f is o-minimal. A function g is definable from f if g is definable in (R,f). Two o-minimal functions are compatible if there exists an o-minimal expansion M of the real field in which they are both definable. I will discuss the o-minimality, the interdefinability and the compatibility of two special functions, Euler’s Gamma and Riemann’s Zeta, restricted to the reals. If time allows it, I will present a general technique for establishing whether a function is definable or not in a given o-minimal expansion of the reals. Joint work with J.-P. Rolin and P. Speissegger.