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  "body": "\n## Definition\n\nLet $(M, g)$ be a closed connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$.\nLet $0 = \\lambda_0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\cdots$ be the [[spectrum of the Laplace–Beltrami operator|B8TYKV7AA1]], with an orthonormal basis of real eigenfunctions $\\phi_0, \\phi_1, \\ldots$ of $L^2(M)$.\nThe *Minakshisundaram–Pleijel zeta function* is\n$$Z(s, x, y) = \\sum_{j \\ge 1} \\frac{\\phi_j(x) \\phi_j(y)}{\\lambda_j^{s}}, \\qquad x, y \\in M,$$\nand its trace on the diagonal is the *spectral zeta function*\n$$Z(s) = \\int_M Z(s, x, x) \\, dv(x) = \\sum_{j \\ge 1} \\lambda_j^{-s}.$$\nBoth series converge for $\\operatorname{Re} s \u003e n/2$.\n\n## Remarks\n\nBy the [[Minakshisundaram–Pleijel expansion|45P2SEGGR1]] of the heat trace and the formula $\\Gamma(s) Z(s) = \\int_0^\\infty t^{s-1} \\big( \\sum_{j \\ge 0} e^{-t\\lambda_j} - 1 \\big) \\, dt$, the function $Z(s)$ has a meromorphic [[continuation|0M3YQJ81NB]] to $\\C$.\nIts poles are simple and lie among the points $s = n/2 - k$, $k = 0, 1, 2, \\ldots$, other than $0, -1, -2, \\ldots$; in particular $Z$ is holomorphic at $s = 0$.\nThe residue at $s = n/2$ is $\\operatorname{vol}(M) / \\big( (4\\pi)^{n/2} \\Gamma(n/2) \\big)$, which is equivalent to Weyl's law.\n\nExample: for $M = \\R / 2\\pi\\Z$ the nonzero eigenvalues are $k^2$, $k \\ge 1$, each with multiplicity $2$.\nSo $Z(s) = 2 \\sum_{k \\ge 1} k^{-2s} = 2 \\zeta(2s)$, with $\\zeta$ the [[Riemann zeta function|1XWBY4N0VT]].\nIt has a single simple pole at $s = 1/2$ with residue $1$, and $Z(0) = 2\\zeta(0) = -1$.\n\nNon-example: the sum $\\sum_{j \\ge 0} \\lambda_j^{-s}$ that includes $\\lambda_0 = 0$ is not defined, since $0^{-s}$ has no meaning; this is why the zero eigenvalue is left out.\n",
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