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