About: Motivic zeta function

In algebraic geometry, the motivic zeta function of a smooth algebraic variety is the formal power series Here is the -th symmetric power of , i.e., the quotient of by the action of the symmetric group , and is the class of in the ring of motives (see below). If the ground field is finite, and one applies the counting measure to , one obtains the local zeta function of . If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to , one obtains .