We prove the existence of an effective power structure over the Grothendieck ring of geometric dg categories. Using this power structure we show that the categorical zeta function of a geometric dg category can be expressed as a power with exponent the category itself. This implies a conjecture of Galkin and Shinder relating the motivic and categorical zeta functions of varieties. We also deduce a formula for the generating series of the classes of derived categories of the Hilbert scheme of points on smooth projective varieties. Moreover, our results show that the Heisenberg action on the derived category of the symmetric orbifold is an irreducible highest weight representation. 查看全文>>