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1. • 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

   $$\beta \mapsto Tr(\exp(- \beta H_{BostConnes}))$$

   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

   $$s \mapsto \int_0^\infty \beta^{s-1} \left(Tr(\exp(- \beta H)) -1\right)\; d \beta \,.$$

   Hm.

2. • CommentRowNumber2.
   • CommentAuthorTom Mainiero
   • CommentTimeSep 11th 2023
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   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