 Let "f" be a function defined on a manifold "M", such that the set of fixed points Fix("f" "n") is finite for all "n" > 1.Further let "φ" be a function on "M" with values in "d" × "d" complex matrices.The zeta function of the first kind is

In the special case "d" = 1, "φ" = 1, we have

which is the Artin–Mazur zeta function.