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High Quality Content by WIKIPEDIA articles! In algebraic geometry, the motivic zeta function of a smooth algebraic variety X is the formal power series Z(X,t)=sum_{n=0}^infty [X^{(n)}]t^n Here X (n) is the n-th symmetric power of X, i.e., the quotient of Xn by the action of the symmetric group Sn, and [X(n)] is the class of X(n) in the ring of motives (see below). If the ground field is finite, and one applies the counting measure to Z(X,t), one obtains the local zeta function of X.