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Relationship between Connes trace formula and Weil's trace formula

Connes trace formula $$ \mathrm{Tr}\,{U(h)}=2h(1)\ln\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|} $$ Weil's trace $$ \int_{C}h(u)|u|d^{*}u- ...