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  "body": "\n## Summary\n\nThe Minakshisundaram–Pleijel zeta function of a closed Riemannian manifold is the Dirichlet series $\\sum \\lambda_j^{-s}$ over the nonzero eigenvalues of its Laplacian.\nIt has a pointwise version, with each term weighted by eigenfunctions.\nIt converges for $\\operatorname{Re} s \u003e n/2$ and continues to a meromorphic function on $\\C$.\nIts poles and special values carry geometric information, and its derivative at $0$ defines the [[zeta-regularized determinant|YT3ZQ2F7FG]] of the Laplacian.\nIt generalizes the [[Riemann zeta function|1XWBY4N0VT]], which arises from the circle.\n\n## Explanations\n\n### L1 human\n\nThe eigenvalues of the Laplacian on a closed manifold are the squared frequencies of its vibrations.\nSumming their negative powers gives a function of $s$, built like the Riemann zeta function from the positive integers.\nFor the circle of length $2\\pi$ the eigenvalues are $1, 1, 4, 4, 9, 9, \\ldots$, and the zeta function is $2\\zeta(2s)$.\nIn general one cannot write it down, but its analytic properties follow from the heat kernel.\n",
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