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nLab > Latest Changes: Bost-Connes system

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeAug 25th 2014
- (edited Aug 25th 2014)
- PermaLink
Author: Urs
Format: MarkdownItexstarted some minimum at _[[Bost-Connes system]]_. Hm, it seems that the statement is that that partition function of the BC-system $$ \beta \mapsto Tr(\exp(- \beta H_{BostConnes})) $$ is the Riemann zeta function. But by the [pertinent analogies](http://ncatlab.org/nlab/show/zeta%20function%20of%20an%20elliptic%20differential%20operator#AnalogyWithNumberTheoreticZetaFunctions) the zeta functions are not supposed to _equal_ partition functions, but to be related to them by the transformation $$ s \mapsto \int_0^\infty \beta^{s-1} \left(Tr(\exp(- \beta H)) -1\right)\; d \beta \,. $$ Hm.

started some minimum at Bost-Connes system.

Hm, it seems that the statement is that that partition function of the BC-system
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>β</mi><mo>↦</mo><mi>Tr</mi><mo stretchy="false">(</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mi>β</mi><msub><mi>H</mi> <mi>BostConnes</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \beta \mapsto Tr(\exp(- \beta H_{BostConnes})) </annotation></semantics></math>$
is the Riemann zeta function. But by the pertinent analogies the zeta functions are not supposed to equal partition functions, but to be related to them by the transformation
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>s</mi><mo>↦</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>∞</mn></msubsup><msup><mi>β</mi> <mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>Tr</mi><mo stretchy="false">(</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mi>β</mi><mi>H</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mspace width="0.27778em"/><mi>d</mi><mi>β</mi><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> s \mapsto \int_0^\infty \beta^{s-1} \left(Tr(\exp(- \beta H)) -1\right)\; d \beta \,. </annotation></semantics></math>$
Hm.
- CommentRowNumber2.
- CommentAuthorTom Mainiero
- CommentTimeSep 11th 2023
- PermaLink
Author: Tom Mainiero
Format: MarkdownItexFixed broken link to the Bost-Connes paper. <a href="https://ncatlab.org/nlab/revision/diff/Bost-Connes+system/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Bost-Connes+system/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Bost-Connes+system">current</a>

Fixed broken link to the Bost-Connes paper.

diff, v4, current

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nLab > Latest Changes: Bost-Connes system