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  "body": "\n## Summary\n\nThe zeta-regularized determinant of a positive elliptic operator is the regularized product of its eigenvalues, $\\exp(-\\zeta'(0))$, where $\\zeta(s) = \\sum \\lambda^{-s}$ is its spectral zeta function.\nRay and Singer introduced it for Laplacians in order to define [[analytic torsion|MZT9D34B0A]].\nIt is an instance of [[zeta function regularization|VJCJYXGTDY]] and appears in string theory, quantum field theory and the geometry of [[determinant line bundles|5DST3VKEMX]].\n\n## Explanations\n\n### L1 human\n\nThe determinant of a matrix is the product of its eigenvalues.\nA differential operator has infinitely many eigenvalues that grow to infinity, so the product diverges.\nThe zeta function trick replaces the logarithm of the product, $\\sum \\log \\lambda$, by $-\\zeta'(0)$, which is finite.\nFor the Laplacian on the circle of length $2\\pi$, with the zero eigenvalue removed, this gives the value $4\\pi^2$.\n",
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