Abstract: If f is a self-map on a set S, consider the sequence of integers f(n) defined as the number of fixed points of the n-th iterate of f (if finite). Patterns in the sequence f(n) can be detected by properties of its associated Artin-Mazur dynamical zeta function. We consider these functions when f is a morphism on an algebraic variety over an algebraically closed field K, and S=X(K). For rational functions on the projective line over the complex numbers, the dynamical zeta function is a rational function, implying a linear recurrence satisfied by f(n). By contrast, in 2012 Andrew Bridy constructed examples of rational functions on the projective line over an algebraically closed field of positive characteristic for which the dynamical zeta function is ?wild?: it is a transcendental function. We shed some light on this phenomenon by defining a "tame zeta function?. In examples, it is an algebraic function, and the ?wild? zeta function is an infinite product of such tame zeta functions. We study in detail the case of endomorphisms of abelian varieties, where we can characterise transcendental zeta function by nilpotent actions on the local p-torsion group scheme. (Joint work with Jakub Byszewski and Lois van der Meijden.)