# Good cohomological setting for binary operations on arithmetical functions

Is there currently a good abstract theory (derived from algebraic geometry and cohomological theories) to study binary operations on arithmetical functions like the Dirichlet convolution $$f\star g = \sum_{d|n} f\left(\frac{n}{d}\right) g(d)\,\, ?$$ I don't know anything in arithmetic geometry, but I know the basic stuff on algebraic geometry (mainly the book of Hartshorne) and I was looking for more advanced cohomological stuff do deal with such operations on arithmetic functions. A collegue told me about crystalline cohomology but was not sure. If it is the case, is there any good book to study theses theories (except SGA) ?

Any helpful comment or book suggestion will be highly appreciated.

• Why do you think that cohomology and algebraic geometry are relevant to Dirichlet convolution? – Daniel Loughran Jan 31 '17 at 14:36
• Deligne's proof of the Weil conjectures allows you to give good bounds on certain exponential sums. Is this what you were thinking about? But these tools are only relevant to very special $f,g$ (e.g. Dirichlet characters or exponentials) – Daniel Loughran Jan 31 '17 at 14:37
• Dirichlet convolution was just an example. I was just wondering if functions defined on $\mathbb{N}$ and operations on these functions can be studied in a cohomological way. I was probably too optimistic but your reference about Deligne's proof is interesting. I will look at that, thank you. – C. Dubussy Jan 31 '17 at 20:31
• Your welcome. Though beware that the technical details involved in translating Deligne's results into explicit bounds for exponential sums can be quite formidable. I'm not sure where one can find an accessible treatment of these; the most famous special case is the so called "Weil bound". – Daniel Loughran Jan 31 '17 at 20:44