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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.

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Category theory is mostly a language to express complicated notions. Your question is akin to, "are there any attempts to prove the Riemann hypothesis in German?" –  Zhen Lin Jun 13 at 12:31
    
@ZhenLin I understand that, my question is "are there any advancements in expressing the Riemann Hypothesis in the language of category theory that might yield useful results." Much in the same way that people often talk about proving an equivalent statement, I was wondering if such an equivalent statement exist in the form of a category theoretic object. Category theory, much like set theory, can be used to formalize things, but still has its own theorems and the like. –  Juan Sebastian Lozano Muñoz Jun 13 at 18:49
    
@ZhenLin I can well imagine that the way how (say) German people think, opens a way towards the Riemann Hypothesis earliest.. –  Berci Jun 14 at 21:32

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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.

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Thank you, this answer is very helpful. 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. –  Juan Sebastian Lozano Muñoz Jun 13 at 18:56

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