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In algebraic geometry, the motivic zeta function of a smooth algebraic variety X {\displaystyle X} is the formal power series
Here X {\displaystyle X^{}} is the n {\displaystyle n} -th symmetric power of X {\displaystyle X} , i.e., the quotient of X n {\displaystyle X^{n}} by the action of the symmetric group S n {\displaystyle S_{n}} , and ] {\displaystyle }]} is the class of X {\displaystyle X^{}} in the ring of motives.
If the ground field is finite, and one applies the counting measure to Z {\displaystyle Z} , one obtains the local zeta function of X {\displaystyle X}.
If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to Z {\displaystyle Z} , one obtains 1 / χ {\displaystyle 1/^{\chi }}.