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.)

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.

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.

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.)