In "On the history of the Weil Conjectures" Dieudonné says: "Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...".
C. F. Gauss, Disquisitiones Arithmeticae; Mathematisches Tagebuch.
A letter to Gauss(1827), C. G. Jacobi.
V. A. Lebesgue work on this type of equations(1837).
Hardy and Littlewood work on the singular series for Waring's problem.
I know that: Hasse proved a conjecture due to Artin about the "Riemann Hypothesis" for curves of genus 1 over Finite fields, Artin conjectured the zeroes of a rational function with coefficients in Q would all be on the circle [u]=q^1/2, and verified that conjecture for many polynomials P of low degree, Weil proved the "Riemann Hypothesis" for a curve of arbitrary genus and for some kind of hypersurfaces and grassmannians among other types of varieties.
My questions are:
What happened between Hardy and Weil work?
Which are the papers of this works?
Is there any book that contains these works?