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.