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  "body": "\n## Definition\n\nLet $M$ be a closed smooth manifold of dimension $n$ with a Hermitian vector bundle $E$.\nLet $A$ be a formally [[self-adjoint|0NEGN5CRSC]] [[elliptic differential operator|MT9P2X85V8]] of order $m \u003e 0$ on $E$, with nonnegative spectrum.\nLet $0 \u003c \\mu_1 \\le \\mu_2 \\le \\cdots$ be its nonzero eigenvalues, counted with multiplicity.\nIts spectral zeta function $\\zeta_A(s) = \\sum_{j \\ge 1} \\mu_j^{-s}$ converges for $\\operatorname{Re} s \u003e n/m$ and has a meromorphic continuation to $\\C$ that is holomorphic at $s = 0$.\nThe *zeta-regularized determinant* of $A$ is\n$$\\det{}'_\\zeta A = \\exp\\big( -\\zeta_A'(0) \\big).$$\nIf $\\ker A = 0$ it is written $\\det_\\zeta A$.\n\n## Remarks\n\nIt is the [[zeta-regularized product|VJCJYXGTDY]] of the nonzero eigenvalues.\nFor the Laplace–Beltrami operator, $\\zeta_A$ is the [[Minakshisundaram–Pleijel zeta function|T1039Q9FY0]].\nIf $A$ were a positive definite matrix, $-\\zeta_A'(0) = \\sum_j \\log \\mu_j$ and the definition would give the usual determinant.\nThe regularized determinant is not multiplicative in general: $\\det_\\zeta(AB)$ may differ from $\\det_\\zeta A \\det_\\zeta B$ by a multiplicative anomaly.\n\nExample: on $S^1 = \\R / 2\\pi\\Z$ let $A = -d^2/d\\theta^2$.\nThe nonzero eigenvalues are $k^2$ with multiplicity $2$, so $\\zeta_A(s) = 2\\zeta(2s)$ and $\\zeta_A'(0) = 4\\zeta'(0) = -2 \\log(2\\pi)$, using $\\zeta'(0) = -\\tfrac12 \\log(2\\pi)$.\nHence $\\det{}'_\\zeta A = 4\\pi^2$.\n\nNon-example: the operator $-d^2/d\\theta^2 - c$ with $0 \u003c c \u003c 1$ on $S^1$ has the negative eigenvalue $-c$.\nIt does not fit the definition, since $\\mu^{-s}$ for $\\mu \u003c 0$ needs a choice of branch of the logarithm.\n",
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