A MacDonald formula for zeta functions of varieties over finite fields

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We provide a formula for the generating series of the zeta function Z(X,t) of symmetric powers Symⁿ X of varieties over finite fields. This realizes Z(X,t) as an exponentiable motivic measure whose associated Kapranov motivic zeta function takes values in W(R) the big Witt ring of R=W(Z). We apply our formula to compute Z(Symⁿ X,t) in a number of explicit cases. Moreover, we show that all λ-ring motivic measures have zeta functions which are exponentiable. In this setting, the formula for Z(X,t) takes the form of a MacDonald formula for the zeta function.