This does not adress the actual question, but might be useful, if somebody wants to trace back the literature as aluded to in another answer.
Peter Roquette has writteen a series of articles entitled
The Riemann hypothesis in characteristic p, its origin and development several parts, total length close to 200 pages. Bibliographic details and pdf are avalaible here: http://www.rzuser.uni-heidelberg.de/~ci3/manu.html (see papers 26,27,36).
In particular, in later parts various of Hasse's paper related to this are discussed.