TITLE: Historical Articles about zeta functions of curves
QUESTION [11 upvotes]: Are there any historical articles about the origins of zeta functions of curves over global fields (undoubtedly starting with $\mathbb{Q}$)?  In particular who (and when did this happen) first have the idea of creating such a thing?  Was it by analogy with the zeta functions of number fields?

REPLY [3 votes]: 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.