A cohomological statement equivalent to the Riemann Hypothesis

It has been speculated for a long time that etale cohomology should have a number field analogue that can be used to state and prove the Weil conjecture for number fields. The cohomology should be infinite dimensional and carry an operator (analogue to Frobenius) and an inner product preserved by the operator. Eigenvalues of the operator of course correspond to zeros of L-functions.

The first published attempt to describe any of this precisely was Deninger's articles on gamma factors and infinite-dimensional (zeta-regularized) determinants. This was at first a purely formal construction of cohomology at the Archimedean places, infinite dimensional and with an operator whose determinant gave the Archimedean factor of the L-function. The neat, classical simplicity of this approach (in the first paper) and its connection to concrete formulas seemed to show there was some hope for making precise statements that could be proved. Deninger continued to refine these analogies in a series of papers over the next 20 years, showing that you could also incorporate the epsilon constants, the Riemann hypothesis for higher-dimensional varieties and motives, and other known or believed features of arithmetic geometry, into a single formalism. In some cases the formalism makes interesting predictions that were proved (by Deninger and his students) and this is evidence that program can eventually succeed. Although somewhat formal I think this is considered very beautiful stuff.

Another step was taken in Katerina Consani's thesis, where she showed that the Archimedean cohomology and its operator (what Deninger had constructed) has a description in terms of the geometry of the variety. This has been greatly generalized and extended in papers by Consani, Marcolli and Connes, and integrated with many of Connes' ideas on the expression of the Riemann hypothesis using noncommutative goemetry. It is also related to the idea that some of the geometry at Archimedian places has a "reduction modulo $\infty$".

A third direction, originating in arithmetic topology (ie analogies to 3-manifolds) and conjectures on values of L-functions, can be found in papers by Baptiste Morin. He shows that if there is a Weil-etale topos as postulated by Lichtenbaum then some aspects of Deninger's program can be carried out.

Another strand is some recent work on integration of Arakelov theory into the framework of motivic cohomology. The state of the art is Jakob Schkolbach's recent papers on the arxiv, on canonical construction of Arakelov cohomology, but this line of work actually goes back to papers by Burgos-Gil and Goncharov on higher Chow groups in Arakelov geometry. This approach does not yet touch questions related to the zeta zeros but is fairly closely related to special values of L-functions.

In case you get the feeling from this enormous body of work that there is an unstoppable momentum toward proving the Riemann Hypothesis in a cohomological framework, Peter Sarnak has published the opinion that construction of an operator on Hilbert space as a way to prove the hypothesis will not work, and that the core of the proof will be the ideas on monodromy of families as in the finite field case.

Christopher Deninger has described the properties of a cohomology theory of algebraic schemes over $\mathrm{Spec}(\mathbb{Z})$ which should be able to solve the Riemann hypothesis, in fact many conjectures about Zeta- and L-functions.

C. Deninger, Some Analogies Between Number Theory and Dynamical Systems on Foliated Spaces, Documenta Mathematica, Extra Volume ICM I (1998), 163-186. online

See also his subsequent papers, for example Arithmetic Geometry and Analysis on Foliated Spaces, online.

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.