TITLE: Modern Algebraic Geometry and Analytic Number Theory
QUESTION [7 upvotes]: I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (with the exception of probabilities). That is to say, to understand it, you really need to know everything. It also has extraordinary opportunities in the understanding of arithmetic (Pierre Deligne in the proofs of André Weil etc.).
However, I don't see any connection with the analytic number theory like the one undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on.

Does anyone have ideas of theorems, conjectures, or "approaches" that
  combine these two points of view?

REPLY [2 votes]: From the point of view of analytic number theory the most important specific result which is proved using algebraic geometry is Burgess' bounds for character sums. The proof relies on Wiles bound for character sums, together with a rather complicated combinatorial argument. One could argue that as Stepanov, Schmidt, and Bombieri gave independent proofs of the required bounds, Weils bounds are not really required, but the "elementary" approach is certainly not easy either.