Dynamical zeta functions and Kummer congruences

We establish a connection between the coefficients of Artin-Mazur zeta-functions and Kummer congruences. This allows to settle positively the question of the existence of a map T : X → X such that the number of fixed points of T n are |E2n|, where E2n are the Euler numbers. Also we solve a problem of Gabcke related to the coefficients of Riemann-Siegel formula.