Geodesic
Zeta Functions in Triangulated Categories
Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 369-381

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We prove the 2-out-of-3 property for the rationality of motivic zeta function in distinguished triangles in Voevodsky's category $\mathscr{DM}$. As an application, we show the rationality of motivic zeta functions for all varieties whose motives are in the thick triangulated monoidal subcategory generated by motives of quasi-projective curves in $\mathscr{DM}$. Together with a result due to P. O'Sullivan, this also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational.
Keywords: zeta function, motivic measure, finite-dimensional motives, triangulated category of motives over a field, homotopy category of motivic symmetric spectra, Grothendieck group of a triangulated category, $\lambda$-ring, rationality. 
@article{MZM_2010_87_3_a4,
     author = {V. I. Guletskii},
     title = {Zeta {Functions} in {Triangulated} {Categories}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {369--381},
     publisher = {mathdoc},
     volume = {87},
     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a4/}
}
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V. I. Guletskii. Zeta Functions in Triangulated Categories. Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 369-381. http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a4/