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What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?
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In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the definition of $T(s)$?
Atiyah states that this function is defined in his paper "The Fine Structure Constant", but I can't seem to find a copy of the paper. So can anyone tell me how Atiyah defined $T$ in that paper?
open-problems definitions riemann-hypothesis
asked 6 hours ago
Keshav Srinivasan
1,101824
 |Â
show 9 more comments
up vote
14
down vote
favorite
4
In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the definition of $T(s)$?
Atiyah states that this function is defined in his paper "The Fine Structure Constant", but I can't seem to find a copy of the paper. So can anyone tell me how Atiyah defined $T$ in that paper?
open-problems definitions riemann-hypothesis
asked 6 hours ago
Keshav Srinivasan
1,101824
8
meta.mathoverflow.net/questions/3894/…
– Mike Miller
6 hours ago
7
Why do people ask why OP wants to know this? Does it matter? It's a good mathematical question.
– Manuel Bärenz
3 hours ago
6
@ManuelBärenz I was trying to avoid discussion, but... T is defined as a composite of isomorphisms $mathbbC stackrelt_+to Z(A) stackrelt_-tomathbbC$ where $Z(A)$ is apparently the centre of the hyperfinite type II von Neumann factor, and each $t_pm$ is induced (somehow, it's not clear) by the map sending a 2x2 complex matrix to its eigenvalues (and recalling that $A$ is an infinite tensor product of such 2x2 matrix algebras). I'm not sure these maps $t_pm$ are well-defined, or if only $T$ is supposed to be, but even then I'm suspicious. I'm not sure it's continuous, even...
– David Roberts
3 hours ago
6
@ManuelBärenz It is difficult to put this diplomatically, but when you express a belief that " I'd still like to learn. It would be incredibly valuable if..." is putting quite a lot of implicit faith in there being something extractable from these documents. Based on my first impressions, I certainly won't be the one to try and do it
– Yemon Choi
3 hours ago
7
@ManuelBärenz I hesitate to say "most constructions" but I can say (perhaps demonstrating less tact than others have been doing) that there are problems even in the explanations of how one is supposed to get started. Here's another instance from the $alpha$ preprint: it is claimed that a finite von Neumann algebras always has a trace (true) and then it is claimed that inner automorphisms give different but isomorphic traces. However, if $tau$ is a trace on any algebra and $phi$ is an inner automorphism then one will find rather quickly that $taucircphi=tau$.
– Yemon Choi
2 hours ago
 |Â
show 9 more comments
up vote
14
down vote
favorite
4
up vote
14
down vote
favorite
4
4
In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the definition of $T(s)$?
Atiyah states that this function is defined in his paper "The Fine Structure Constant", but I can't seem to find a copy of the paper. So can anyone tell me how Atiyah defined $T$ in that paper?
open-problems definitions riemann-hypothesis
asked 6 hours ago
Keshav Srinivasan
1,101824
In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the definition of $T(s)$?
Atiyah states that this function is defined in his paper "The Fine Structure Constant", but I can't seem to find a copy of the paper. So can anyone tell me how Atiyah defined $T$ in that paper?
open-problems definitions riemann-hypothesis
open-problems definitions riemann-hypothesis
asked 6 hours ago
Keshav Srinivasan
1,101824
asked 6 hours ago
Keshav Srinivasan
1,101824
asked 6 hours ago
Keshav Srinivasan
1,101824
asked 6 hours ago
Keshav Srinivasan
1,101824
asked 6 hours ago
Keshav Srinivasan
1,101824
1,101824
8
meta.mathoverflow.net/questions/3894/…
– Mike Miller
6 hours ago
7
Why do people ask why OP wants to know this? Does it matter? It's a good mathematical question.
– Manuel Bärenz
3 hours ago
6
@ManuelBärenz I was trying to avoid discussion, but... T is defined as a composite of isomorphisms $mathbbC stackrelt_+to Z(A) stackrelt_-tomathbbC$ where $Z(A)$ is apparently the centre of the hyperfinite type II von Neumann factor, and each $t_pm$ is induced (somehow, it's not clear) by the map sending a 2x2 complex matrix to its eigenvalues (and recalling that $A$ is an infinite tensor product of such 2x2 matrix algebras). I'm not sure these maps $t_pm$ are well-defined, or if only $T$ is supposed to be, but even then I'm suspicious. I'm not sure it's continuous, even...
– David Roberts
3 hours ago
6
@ManuelBärenz It is difficult to put this diplomatically, but when you express a belief that " I'd still like to learn. It would be incredibly valuable if..." is putting quite a lot of implicit faith in there being something extractable from these documents. Based on my first impressions, I certainly won't be the one to try and do it
– Yemon Choi
3 hours ago
7
@ManuelBärenz I hesitate to say "most constructions" but I can say (perhaps demonstrating less tact than others have been doing) that there are problems even in the explanations of how one is supposed to get started. Here's another instance from the $alpha$ preprint: it is claimed that a finite von Neumann algebras always has a trace (true) and then it is claimed that inner automorphisms give different but isomorphic traces. However, if $tau$ is a trace on any algebra and $phi$ is an inner automorphism then one will find rather quickly that $taucircphi=tau$.
– Yemon Choi
2 hours ago
 |Â
show 9 more comments
8
meta.mathoverflow.net/questions/3894/…
– Mike Miller
6 hours ago
7
Why do people ask why OP wants to know this? Does it matter? It's a good mathematical question.
– Manuel Bärenz
3 hours ago
6
@ManuelBärenz I was trying to avoid discussion, but... T is defined as a composite of isomorphisms $mathbbC stackrelt_+to Z(A) stackrelt_-tomathbbC$ where $Z(A)$ is apparently the centre of the hyperfinite type II von Neumann factor, and each $t_pm$ is induced (somehow, it's not clear) by the map sending a 2x2 complex matrix to its eigenvalues (and recalling that $A$ is an infinite tensor product of such 2x2 matrix algebras). I'm not sure these maps $t_pm$ are well-defined, or if only $T$ is supposed to be, but even then I'm suspicious. I'm not sure it's continuous, even...
– David Roberts
3 hours ago
6
@ManuelBärenz It is difficult to put this diplomatically, but when you express a belief that " I'd still like to learn. It would be incredibly valuable if..." is putting quite a lot of implicit faith in there being something extractable from these documents. Based on my first impressions, I certainly won't be the one to try and do it
– Yemon Choi
3 hours ago
7
@ManuelBärenz I hesitate to say "most constructions" but I can say (perhaps demonstrating less tact than others have been doing) that there are problems even in the explanations of how one is supposed to get started. Here's another instance from the $alpha$ preprint: it is claimed that a finite von Neumann algebras always has a trace (true) and then it is claimed that inner automorphisms give different but isomorphic traces. However, if $tau$ is a trace on any algebra and $phi$ is an inner automorphism then one will find rather quickly that $taucircphi=tau$.
– Yemon Choi
2 hours ago
8
8
meta.mathoverflow.net/questions/3894/…
– Mike Miller
6 hours ago
meta.mathoverflow.net/questions/3894/…
– Mike Miller
6 hours ago
7
7
Why do people ask why OP wants to know this? Does it matter? It's a good mathematical question.
– Manuel Bärenz
3 hours ago
Why do people ask why OP wants to know this? Does it matter? It's a good mathematical question.
– Manuel Bärenz
3 hours ago
6
6
@ManuelBärenz I was trying to avoid discussion, but... T is defined as a composite of isomorphisms $mathbbC stackrelt_+to Z(A) stackrelt_-tomathbbC$ where $Z(A)$ is apparently the centre of the hyperfinite type II von Neumann factor, and each $t_pm$ is induced (somehow, it's not clear) by the map sending a 2x2 complex matrix to its eigenvalues (and recalling that $A$ is an infinite tensor product of such 2x2 matrix algebras). I'm not sure these maps $t_pm$ are well-defined, or if only $T$ is supposed to be, but even then I'm suspicious. I'm not sure it's continuous, even...
– David Roberts
3 hours ago
@ManuelBärenz I was trying to avoid discussion, but... T is defined as a composite of isomorphisms $mathbbC stackrelt_+to Z(A) stackrelt_-tomathbbC$ where $Z(A)$ is apparently the centre of the hyperfinite type II von Neumann factor, and each $t_pm$ is induced (somehow, it's not clear) by the map sending a 2x2 complex matrix to its eigenvalues (and recalling that $A$ is an infinite tensor product of such 2x2 matrix algebras). I'm not sure these maps $t_pm$ are well-defined, or if only $T$ is supposed to be, but even then I'm suspicious. I'm not sure it's continuous, even...
– David Roberts
3 hours ago
6
6
@ManuelBärenz It is difficult to put this diplomatically, but when you express a belief that " I'd still like to learn. It would be incredibly valuable if..." is putting quite a lot of implicit faith in there being something extractable from these documents. Based on my first impressions, I certainly won't be the one to try and do it
– Yemon Choi
3 hours ago
@ManuelBärenz It is difficult to put this diplomatically, but when you express a belief that " I'd still like to learn. It would be incredibly valuable if..." is putting quite a lot of implicit faith in there being something extractable from these documents. Based on my first impressions, I certainly won't be the one to try and do it
– Yemon Choi
3 hours ago
7
7
@ManuelBärenz I hesitate to say "most constructions" but I can say (perhaps demonstrating less tact than others have been doing) that there are problems even in the explanations of how one is supposed to get started. Here's another instance from the $alpha$ preprint: it is claimed that a finite von Neumann algebras always has a trace (true) and then it is claimed that inner automorphisms give different but isomorphic traces. However, if $tau$ is a trace on any algebra and $phi$ is an inner automorphism then one will find rather quickly that $taucircphi=tau$.
– Yemon Choi
2 hours ago
@ManuelBärenz I hesitate to say "most constructions" but I can say (perhaps demonstrating less tact than others have been doing) that there are problems even in the explanations of how one is supposed to get started. Here's another instance from the $alpha$ preprint: it is claimed that a finite von Neumann algebras always has a trace (true) and then it is claimed that inner automorphisms give different but isomorphic traces. However, if $tau$ is a trace on any algebra and $phi$ is an inner automorphism then one will find rather quickly that $taucircphi=tau$.
– Yemon Choi
2 hours ago
 |Â
show 9 more comments
1 Answer
1
active
oldest
votes
up vote
2
down vote
Here's a public paper of the "the fine structure constant" by Atiyah.
It doesn't seem to be the original, but a copy:
https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view
See the section 3.4, the Todd function is defined there.
answered 17 mins ago
Wilem2
663
1
you might want to check the comments by David Roberts in the OP for why this is not really a "definition"
– Carlo Beenakker
10 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Here's a public paper of the "the fine structure constant" by Atiyah.
It doesn't seem to be the original, but a copy:
https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view
See the section 3.4, the Todd function is defined there.
answered 17 mins ago
Wilem2
663
1
you might want to check the comments by David Roberts in the OP for why this is not really a "definition"
– Carlo Beenakker
10 mins ago
add a comment |Â
up vote
2
down vote
Here's a public paper of the "the fine structure constant" by Atiyah.
It doesn't seem to be the original, but a copy:
https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view
See the section 3.4, the Todd function is defined there.
answered 17 mins ago
Wilem2
663
1
you might want to check the comments by David Roberts in the OP for why this is not really a "definition"
– Carlo Beenakker
10 mins ago
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Here's a public paper of the "the fine structure constant" by Atiyah.
It doesn't seem to be the original, but a copy:
https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view
See the section 3.4, the Todd function is defined there.
answered 17 mins ago
Wilem2
663
Here's a public paper of the "the fine structure constant" by Atiyah.
It doesn't seem to be the original, but a copy:
https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view
See the section 3.4, the Todd function is defined there.
answered 17 mins ago
Wilem2
663
answered 17 mins ago
Wilem2
663
answered 17 mins ago
Wilem2
663
answered 17 mins ago
Wilem2
663
663
1
you might want to check the comments by David Roberts in the OP for why this is not really a "definition"
– Carlo Beenakker
10 mins ago
add a comment |Â
1
you might want to check the comments by David Roberts in the OP for why this is not really a "definition"
– Carlo Beenakker
10 mins ago
1
1
you might want to check the comments by David Roberts in the OP for why this is not really a "definition"
– Carlo Beenakker
10 mins ago
you might want to check the comments by David Roberts in the OP for why this is not really a "definition"
– Carlo Beenakker
10 mins ago
add a comment |Â
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8
meta.mathoverflow.net/questions/3894/…
– Mike Miller
6 hours ago
7
Why do people ask why OP wants to know this? Does it matter? It's a good mathematical question.
– Manuel Bärenz
3 hours ago
6
@ManuelBärenz I was trying to avoid discussion, but... T is defined as a composite of isomorphisms $mathbbC stackrelt_+to Z(A) stackrelt_-tomathbbC$ where $Z(A)$ is apparently the centre of the hyperfinite type II von Neumann factor, and each $t_pm$ is induced (somehow, it's not clear) by the map sending a 2x2 complex matrix to its eigenvalues (and recalling that $A$ is an infinite tensor product of such 2x2 matrix algebras). I'm not sure these maps $t_pm$ are well-defined, or if only $T$ is supposed to be, but even then I'm suspicious. I'm not sure it's continuous, even...
– David Roberts
3 hours ago
6
@ManuelBärenz It is difficult to put this diplomatically, but when you express a belief that " I'd still like to learn. It would be incredibly valuable if..." is putting quite a lot of implicit faith in there being something extractable from these documents. Based on my first impressions, I certainly won't be the one to try and do it
– Yemon Choi
3 hours ago
7
@ManuelBärenz I hesitate to say "most constructions" but I can say (perhaps demonstrating less tact than others have been doing) that there are problems even in the explanations of how one is supposed to get started. Here's another instance from the $alpha$ preprint: it is claimed that a finite von Neumann algebras always has a trace (true) and then it is claimed that inner automorphisms give different but isomorphic traces. However, if $tau$ is a trace on any algebra and $phi$ is an inner automorphism then one will find rather quickly that $taucircphi=tau$.
– Yemon Choi
2 hours ago