Motivic zeta function
In algebraic geometry, the motivic zeta function of a smooth algebraic variety formula_1 is the formal power series Here formula_3 is the formula_4-th symmetric power of formula_1, i.e., the quotient of formula_6 by the action of the symmetric group formula_7, and formula_8 is the class of formula_3 in the ring of motives (see below). If the ground field is finite, and one applies the counting measure to formula_10, one obtains the local zeta function of formula_1. If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to formula_10, one obtains  formula_13. Motivic measures. A motivic measure is a map formula_14 from the set of finite type schemes over a field formula_15 to a commutative ring formula_16, satisfying the three properties For example if formula_15 is a finite field and formula_24 is the ring of integers, then formula_25 defines a motivic measure, the "counting measure". If the ground field is the complex numbers, then Euler characteristic with compact supports defines a motivic measure with values in the integers. The zeta function with respect to a motivic measure formula_14 is the formal power series in formula_27 given by There is a "universal  motivic measure". It takes values in the K-ring of varieties, formula_29, which is the ring generated by the symbols formula_30, for all varieties formula_1, subject to the relations The universal motivic measure gives rise to the motivic zeta function. Examples. Let formula_39 denote the class of the affine line. If formula_1 is a smooth projective irreducible curve of genus formula_44 admitting a line bundle of degree 1, and the motivic measure takes values in a field in which formula_45 is invertible, then where formula_47 is a polynomial of degree formula_48. Thus, in this case, the motivic zeta function is  rational. In higher dimension, the motivic zeta function is not always rational. If formula_49 is a smooth surface over an algebraically closed field of characteristic formula_50, then the generating function for the motives of the Hilbert schemes of formula_49 can be expressed in terms of the motivic zeta function by "Göttsche's Formula" Here formula_53 is the Hilbert scheme of length formula_4 subschemes of formula_49. For the affine plane this formula gives This is essentially the partition function.