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There is a very interesting book which appeared recently precisely on this topic: "The Riemann Hypothesis for Function Fields" by Machiel van Frankenhuijsen. His attempt at answering the OP's question about what is missing is in the last two chapters.

My impression is that Connes' work on RH, like that of great painters such as Picasso, can be divided into different periods. I would say the first period is about exploring connections to statistical mechanics and the notion of symmetry breaking. The main result is his article with Bost on the now called Bost-Connes system. However, this did not yet contain an approach towards RH. The second period is that of his Selecta paper with the trace formula. Finally the third period involves having "fun with $\mathbb{F}_1$", hyperreals, toposes, etc. The third period is about constructing suitable objects which would allow one to mimic Weil's proof. I am not familiar with this last development, but I think Will did a good job at explaining what is missing regarding the approach from this third period. The book I mentioned and the little I know concern the second period.

Let me first give a general somewhat vague opinion: if one has a reasonably clear description of what $A$ and $B$ are, then it is easier to prove a conjecture which says $A=B$ than one which says $A\neq B$ (or $A>B$, $A\ge B$, etc.). For example, I think the recent work by Kronheimer and Mrowka is significant progress on the four-color theorem. The latter says $A>B$ where $B=0$ and $A$ is the number of Tait colorings. They produced some invariant $C$ which they showed satisfies $C>B$ and reduced the problem to the conjecture $A=C$. Connes' Selecta article reduced Weil positivity (an $A\ge B$ type problem) and thus RH to an $A=B$ type problem, i.e., proving his global trace formula which is formally similar to a theorem of Guillemin and Sternberg for flows on Riemannian manifolds. He also proved the semi-local analogue of his formula. Take a finite group $G$ with a unitary representation $U$ and consider the matrix ${\rm tr}(U(gh^{-1}))$ with rows and columns indexed by $g,h\in G$. This matrix is positive semidefinite. Weil positivity follows from Connes' global trace formula for essentially the same reason.

Proving the trace formula, prima facie, looks like a problem of harmonic analysis or even constructive quantum field theory since one has a formally precise but still physicsy looking infinite-dimensional trace to make sense of by introducing UV and IR cutoffs and then analyzing the limit of removing them. There has been follow-ups by Haran (here and here) and Burnol (here and here) but, as far as I know, the trace formula has not been proved in this approach, even in the function field case where at least we know it should be true.

There is a very interesting book which appeared recently precisely on this topic: "The Riemann Hypothesis for Function Fields" by Machiel van Frankenhuijsen. His attempt at answering the OP's question about what is missing is in the last two chapters.

My impression is that Connes' work on RH, like that of great painters such as Picasso, can be divided into different periods. I would say the first period is about exploring connections to statistical mechanics and the notion of symmetry breaking. The main result is his article with Bost on the now called Bost-Connes system. However, this did not yet contain an approach towards RH. The second period is that of his Selecta paper with the trace formula. Finally the third period involves having "fun with $\mathbb{F}_1$", hyperreals, toposes, etc. The third period is about constructing suitable objects which would allow one to mimic Weil's proof. I am not familiar with this last development, but I think Will did a good job at explaining what is missing regarding the approach from this third period. The book I mentioned and the little I know concern the second period.

Let me first give a general somewhat vague opinion: if one has a reasonably clear description of what $A$ and $B$ are, then it is easier to prove a conjecture which says $A=B$ than one which says $A\neq B$ (or $A>B$, $A\ge B$, etc.). For example, I think the recent work by Kronheimer and Mrowka is significant progress on the four-color theorem. The latter says $A>B$ where $B=0$ and $A$ is the number of Tait colorings. They produced some invariant $C$ which they showed satisfies $C>B$ and reduced the problem to the conjecture $A=C$. Connes' Selecta article reduced Weil positivity (an $A\ge B$ type problem) and thus RH to an $A=B$ type problem, i.e., proving his global trace formula which is formally similar to a theorem of Guillemin and Sternberg for flows on Riemannian manifolds. He also proved the semi-local analogue of his formula. Take a finite group $G$ with a unitary representation $U$ and consider the matrix ${\rm tr}(U(gh^{-1}))$ with rows and columns indexed by $g,h\in G$. This matrix is positive semidefinite. Weil positivity follows from Connes' global trace formula for essentially the same reason.

Proving the trace formula, prima facie, looks like a problem of harmonic analysis or even constructive quantum field theory since one has a formally precise but still physicsy looking infinite-dimensional trace to make sense of by introducing UV and IR cutoffs and then analyzing the limit of removing them. There has been follow-ups by Haran (here and here) and Burnol (here and here) but, as far as I know, the trace formula has not been proved in this approach, even in the function field case where at least we know it should be true.

There is a very interesting book which appeared recently precisely on this topic: "The Riemann Hypothesis for Function Fields" by Machiel van Frankenhuijsen. His attempt at answering the OP's question about what is missing is in the last two chapters.

My impression is that Connes' work on RH, like that of great painters such as Picasso, can be divided into different periods. I would say the first period is about exploring connections to statistical mechanics and the notion of symmetry breaking. The main result is his article with Bost on the now called Bost-Connes system. However, this did not yet contain an approach towards RH. The second period is that of his Selecta paper with the trace formula. Finally the third period involves having "fun with $\mathbb{F}_1$", hyperreals, toposes, etc. The third period is about constructing suitable objects which would allow one to mimic Weil's proof. I am not familiar with this last development, but I think Will did a good job at explaining what is missing regarding the approach from this third period. The book I mentioned and the little I know concern the second period.

Let me first give a general somewhat vague opinion: if one has a reasonably clear description of what $A$ and $B$ are, then it is easier to prove a conjecture which says $A=B$ than one which says $A\neq B$ (or $A>B$, $A\ge B$, etc.). For example, I think the recent work by Kronheimer and Mrowka is significant progress on the four-color theorem. The latter says $A>B$ where $B=0$ and $A$ is the number of Tait colorings. They produced some invariant $C$ which they showed satisfies $C>B$ and reduced the problem to the conjecture $A=C$. Connes' Selecta article reduced Weil positivity (an $A\ge B$ type problem) and thus RH to an $A=B$ type problem, i.e., proving his global trace formula which is formally similar to a theorem of Guillemin and Sternberg for flows on Riemannian manifolds. He also proved the semi-local analogue of his formula. Take a finite group $G$ with a unitary representation $U$ and consider the matrix ${\rm tr}(U(gh^{-1}))$ with rows and columns indexed by $g,h\in G$. This matrix is positive semidefinite. Weil positivity follows from Connes' global trace formula for essentially the same reason.

Proving the trace formula, prima facie, looks like a problem of harmonic analysis or even constructive quantum field theory since one has a formally precise but still physicsy looking infinite-dimensional trace to make sense of by introducing UV and IR cutoffs and then analyzing the limit of removing them. There has been follow-ups by Haran (here and here) and Burnol (here and here) but, as far as I know, the trace formula has not been proved in this approach, even in the function field case where at least we know it should be true.

There is a very interesting book which appeared recently precisely on this topic: "The Riemann Hypothesis for Function Fields" by Machiel van Frankenhuijsen. His attempt at answering the OP's question about what is missing is in the last two chapters.

My impression is that Connes' work on RH, like that of great painters such as Picasso, can be divided into different periods. I would say the first period is about exploring connections to statistical mechanics and the notion of symmetry breaking. The main result is his article with Bost on the now called Bost-Connes system. However, this did not yet contain an approach towards RH. The second period is that of his Selecta paper with the trace formula. Finally the third period involves having "fun with $\mathbb{F}_1$", hyperreals, toposes, etc. The third period is about constructing suitable objects which would allow one to mimic Weil's proof. I am not familiar with this last development, but I think Will did a good job at explaining what is missing regarding the approach from this third period. The book I mentioned and the little I know concern the second period.

Let me first give a general somewhat vague opinion: if one has a reasonably clear description of what $A$ and $B$ are, then it is easier to prove a conjecture which says $A=B$ than one which says $A\neq B$ (or $A>B$, $A\ge B$, etc.). For example, I think the recent work by Kronheimer and Mrowka is significant progress on the four-color theorem. The latter says $A>B$ where $B=0$ and $A$ is the number of Tait colorings. They produced some invariant $C$ which they showed satisfies $C>B$ and reduced the problem to the conjecture $A=C$. Connes' Selecta article reduced Weil positivity (an $A\ge B$ type problem) and thus RH to an $A=B$ type problem, i.e., proving his global trace formula which is formally similar to a theorem of Guillemin and Sternberg for flows on Riemannian manifolds. He also proved the semi-local analogue of his formula. Take a finite group $G$ with a unitary representation $U$ and consider the matrix ${\rm tr}(U(gh^{-1}))$ with rows and columns indexed by $g,h\in G$. This matrix is positive semidefinite. Weil positivity follows from Connes' global trace formula for essentially the same reason.

Proving the trace formula, prima facie, looks like a problem of harmonic analysis or even constructive quantum field theory since one has a formally precise but still physicsy looking infinite-dimensional trace to make sense of by introducing UV and IR cutoffs and then analyzing the limit of removing them. There has been follow-ups by Haran (here and here) and Burnol (here and here) but, as far as I know, the trace formula has not been proved in this approach, even in the function field case where at least we know it should be true.

There is a very interesting book which appeared recently precisely on this topic: "The Riemann Hypothesis for Function Fields" by Machiel van Frankenhuijsen. His attempt at answering the OP's question about what is missing is in the last two chapters.

My impression is that Connes' work on RH, like that of great painters such as Picasso, can be divided into different periods. I would say the first period is about exploring connections to statistical mechanics and the notion of symmetry breaking. The main result is his article with Bost on the now called Bost-Connes system. However, this did not yet contain an approach towards RH. The second period is that of his Selecta paper with the trace formula. Finally the third period involves having "fun with $\mathbb{F}_1$", hyperreals, toposes, etc. The third period is about constructing suitable objects which would allow one to mimic Weil's proof. I am not familiar with this last development, but I think Will did a good job at explaining what is missing regarding the approach from this third period. The book I mentioned and the little I know concern the second period.

Let me first give a general somewhat vague opinion: if one has a reasonably clear description of what $A$ and $B$ are, then it is easier to prove a conjecture which says $A=B$ than one which says $A\neq B$ (or $A>B$, $A\ge B$, etc.). For example, I think the recent work by Kronheimer and Mrowka is significant progress on the four-color theorem. The latter says $A>B$ where $B=0$ and $A$ is the number of Tait colorings. They produced some invariant $C$ which they showed satisfies $C>B$ and reduced the problem to the conjecture $A=C$. Connes' Selecta article reduced Weil positivity (an $A\ge B$ type problem) and thus RH to an $A=B$ type problem, i.e., proving his global trace formula which is formally similar to a theorem of Guillemin and Sternberg for flows on Riemannian manifolds. He also proved the semi-local analogue of his formula. Take a finite group $G$ with unitary representation $U$ and consider the matrix ${\rm tr}(U(gh^{-1}))$ with rows and columns indexed by $g,h\in G$. This matrix is positive semidefinite. Weil positivity follows from Connes' global trace formula for essentially the same reason.

Proving the trace formula, prima facie, looks like a problem of harmonic analysis or even constructive quantum field theory since one has a formally precise but still physicsy looking infinite-dimensional trace to make sense of by introducing UV and IR cutoffs and then analyzing the limit of removing them. There has been follow-ups by Haran (here and here) and Burnol (here and here) but, as far as I know the, trace formula has not been proved in this approach, even in the function field case where at least we know it should be true.

There is a very interesting book which appeared recently precisely on this topic: "The Riemann Hypothesis for Function Fields" by Machiel van Frankenhuijsen. His attempt at answering the OP's question about what is missing is in the last two chapters.

My impression is that Connes' work on RH, like that of great painters such as Picasso, can be divided into different periods. I would say the first period is about exploring connections to statistical mechanics and the notion of symmetry breaking. The main result is his article with Bost on the now called Bost-Connes system. However, this did not yet contain an approach towards RH. The second period is that of his Selecta paper with the trace formula. Finally the third period involves having "fun with $\mathbb{F}_1$", hyperreals, toposes, etc. The third period is about constructing suitable objects which would allow one to mimic Weil's proof. I am not familiar with this last development, but I think Will did a good job at explaining what is missing regarding the approach from this third period. The book I mentioned and the little I know concern the second period.

Let me first give a general somewhat vague opinion: if one has a reasonably clear description of what $A$ and $B$ are, then it is easier to prove a conjecture which says $A=B$ than one which says $A\neq B$ (or $A>B$, $A\ge B$, etc.). For example, I think the recent work by Kronheimer and Mrowka is significant progress on the four-color theorem. The latter says $A>B$ where $B=0$ and $A$ is the number of Tait colorings. They produced some invariant $C$ which they showed satisfies $C>B$ and reduced the problem to the conjecture $A=C$. Connes' Selecta article reduced Weil positivity (an $A\ge B$ type problem) and thus RH to an $A=B$ type problem, i.e., proving his global trace formula which is formally similar to a theorem of Guillemin and Sternberg for flows on Riemannian manifolds. He also proved the semi-local analogue of his formula. Take a finite group $G$ with a unitary representation $U$ and consider the matrix ${\rm tr}(U(gh^{-1}))$ with rows and columns indexed by $g,h\in G$. This matrix is positive semidefinite. Weil positivity follows from Connes' global trace formula for essentially the same reason.

Proving the trace formula, prima facie, looks like a problem of harmonic analysis or even constructive quantum field theory since one has a formally precise but still physicsy looking infinite-dimensional trace to make sense of by introducing UV and IR cutoffs and then analyzing the limit of removing them. There has been follow-ups by Haran (here and here) and Burnol (here and here) but, as far as I know, the trace formula has not been proved in this approach, even in the function field case where at least we know it should be true.