On twisted A-harmonic sums and Carlitz finite zeta values

Twisted A-harmonic sums are partial sums of a class of zeta values introduced by the first author. We prove some new identities for such sums and we deduce properties of analogues of finite zeta values in the framework of the Carlitz module. In the theory of finite multiple zeta values as introduced by Kaneko and Zagier, finite zeta values are all zero and there is no known non-zero finite multiple zeta value. In the Carlitzian setting the phenomenology is different as we can deduce, from our results, the irrationality of certain finite zeta values.