I was wondering if there has been any category theoretic advancements in the study of the Riemann Hypothesis and the theory surrounding it?
This question is meant in the same vein as these questions, but specifically for category theory. These other two questions do not have any answers related specifically to category theory.
EDIT: Just to be clear, and because Zhen Lin brought up a good point about category theory being mostly a language for mathematics: I understand that category theory is a language like thing, but it can still be useful in mathematics. I had in mind maybe an advancement in one of the fields related to the RH, in which category theory plays an important or central role. Or perhaps a category theoretic equivalent statement or something similar.
There have been many category theoretic advancements, e.g., in the work on Deligne's "Weil II version" of the Riemann Hypothesis over finite fields. A reference is the book "Convolution and Equidistribution Sato-Tate Theorems for Finite-field Mellin Transforms" by Nicholas M. Katz.
In general, category theory has advanced many fields, like noncommutative algebraic geometry, homotopical algebra, homological algebra, topological field theory, and others.
Wikipedia lists possible attempts to prove RH, see here, but category theory is not explictily mentioned (although it is present in some of the fields, as mentioned above). It seems to me that there is no special category theoretical approach to RH like you had perhaps in mind.