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).

I am not familiar with the content of these paper, the following is based on a very rough browsing. 

The covered time span seems to be roughly 1920-40. 

E. Artin's (1921) and F.K. Schmidt's (1925) theses seem the most important starting points. 
However, several other names are mentioned as well; H. Hasse worked on the Riemann  Hyothesis (in this context) around 33/34; referring (freely translated) to Artin and Schmidt  zeta-functions.  

F.K. Schmidt: Zur Zahlentheorie in Koorpern der Charakteristik p. (Vorlaeufige Mitteilung.)
Sitz.-Ber. phys. med. Soz. Erlangen 58/59 (1926/27) 159–172

is mentioned as the paper introducing the zeta-function for function fields (fin.gen. and trans. deg. 1). 

As motivation for these investigations, the analogy with the number field case is mentioned.


ADDED: In retrospect, I am now unsure whether this is what was asked for.
Sorry, in case it should not be. (Zhich in view of a comment, appearing while I added the disclaimer, seems to be the case.)