TITLE: A cohomological statement equivalent to the Riemann Hypothesis
QUESTION [11 upvotes]: Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?

REPLY [3 votes]: Sarnak's opinion that "the proof will be the ideas on monodromy of families as in the finite field case."  can be found in a paper named (in hints) "Problems of the Millennium: The Riemann Hypothesis" in this book "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", and expanded in the book (he co authored it) "Random Matrices,
Frobenius Eigenvalues,and Monodromy, by Nicholas M. Katz,Peter Sarnak". Hope this information helps.