Irrationality of the motivic zeta function

Kapranov defined a motivic zeta function $\zeta(X)$ of a variety $X$ as a power series with coefficients in the Grothendieck ring of varieties. It is a motivic lift of the classical Hasse-Weil zeta function of $X$. Kapranov proved that $\zeta(X)$ is rational if $\dim X$ is at most one. He then also conjectured that rationality holds for all varieties $X$. Jointly with Michael Larsen we disproved this conjecture in the paper math/0110255. However our method does not prove irrationality if one considers the localization of the Grothendieck ring of varieties by inverting $\mathbb{L}$ – the class of the affine line. Actually Denef and Loeser conjectured that $\zeta(X)$ is rational in this localized ring. I will report on our recent theorem with Larsen which claims irrationality even after inverting $\mathbb{L}$.