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Let $\\Lambda$ be the Dirichlet-to-Neumann map on $\\Gamma$ and let ${\\rm det}_\\zeta(\\Lambda)$ be its (modified, i. e. with zero mode excluded) $\\zeta$-regularized determinant. It is well-known that the quantity ${\\rm det}_\\zeta(\\Lambda)/|\\Gamma|$ (where $|\\Gamma|$ is the length of $\\Gamma$) is a conformal invariant. 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Let $\\Lambda$ be the Dirichlet-to-Neumann map on $\\Gamma$ and let ${\\rm det}_\\zeta(\\Lambda)$ be its (modified, i. e. with zero mode excluded) $\\zeta$-regularized determinant. It is well-known that the quantity ${\\rm det}_\\zeta(\\Lambda)/|\\Gamma|$ (where $|\\Gamma|$ is the length of $\\Gamma$) is a conformal invariant. 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