We study the natural λ-ring structure on the Grothendieck ring of the triangulated category of mixed motives. Basic properties of a natural notion of characteristic-like series are developed in the context of equivariant objects.

Let C be a pseudo-abelian symmetric monoidal category, and X a Schur-finite object of C. We study the problem of rationality of the motivic zeta function ζx(t) of X. Since the coefficient ring is not a field, there are several variants of rationality — uniform, global, determinantal and pointwise rationality. We show that ζx(t) is determinantally rational, and we give an example of C and X for which the motivic zeta function is not uniformly rational.

The zeta function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta function of a surface is rational if and only if the Kodaira dimension of the surface is negative.

Let $(S, 0)$ be a normal surface germ and Let $f$ a non-constant regular function on Let $(S, 0)$ with Let $f(0) = 0$. Using any additive invariant on complex algebraic varieties one can associate a zeta function to these data, where the topological and motivic zeta functions are the roughest and the finest zeta functions, respectively. In this paper we are interested in a geometric determination of the poles of these functions. The second author has already provided such a determination for the topological zeta function in the case of non-singular surfaces. Here we give a complete answer for all normal surfaces, at least on the motivic level. The topological zeta function however seems to be too rough for this purpose, although for negative poles, which are the only ones in the non-singular case, we are able to prove exactly the same result as for non-singular surfaces.

We also give and verify a (natural) definition for when a rational number is a pole of the motivic zeta function.

Let $X$ be a nonsingular algebraic variety in characteristic zero. To an effective divisor on $X$ Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on $X$, which specializes to both the classical $p$-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.

This paper treats a generalization to singular varieties. Batyrev already considered such a ‘Kontsevich invariant’ for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any $\mathbb{Q}$-Gorenstein variety $X$ we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective $\mathbb{Q}$-Cartier divisors on $X$ whose support contains the singular locus of $X$.