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A theorem of Tate and Turner says that global function fields have the samezeta function if and only if the Jacobians of the corresponding curves areisogenous. In this note, we investigate what happens if we replace the usual(characteristic zero) zeta function by the positive characteristic zetafunction introduced by Goss. We prove that for function fields whosecharacteristic exceeds their degree, equality of the Goss zeta function is thesame as Gassmann-equivalence (a purely group theoretical property), but thisstatement fails if the degree exceeds the characteristic. We introduce a`Teichmueller lift' of the Goss zeta function and show that equality of such isalways the same as Gassmann equivalence.

Arithmetic equivalence for function fields, the Goss zeta function and a generalization