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  "body": "\n## Definition\n\nLet $Y$ be a closed smooth manifold of dimension $n$ with a smooth positive density, and let $E$ be a Hermitian vector bundle over $Y$.\nLet $A$ be a formally [[self-adjoint|0NEGN5CRSC]] [[elliptic differential operator|MT9P2X85V8]] of order $m \\ge 1$ on sections of $E$.\nIts spectrum is a discrete set of real [[eigenvalues|0BQAA2BK3Q]] $\\lambda$ of finite multiplicity.\nThe *eta function* of $A$ is\n$$\\eta_A(s) = \\sum_{\\lambda \\ne 0} \\operatorname{sign}(\\lambda) \\, \\abs{\\lambda}^{-s},$$\nsummed over the nonzero eigenvalues counted with multiplicity.\nIt converges for $\\operatorname{Re} s \u003e n / m$, and it has a meromorphic [[continuation|0M3YQJ81NB]] to $\\C$ that is holomorphic at $s = 0$.\nThe *eta invariant* of $A$ is $\\eta_A(0)$.\n\n## Remarks\n\nThe regularity at $s = 0$ was proved by Atiyah, Patodi and Singer, with the general case due to Gilkey.\nSome authors call $\\xi_A = \\tfrac12(\\eta_A(0) + \\dim \\ker A)$ the reduced eta invariant.\n\nExample: on $S^1 = \\R / 2\\pi\\Z$ let $A = -i \\, d/d\\theta + a$ with $0 \u003c a \u003c 1$.\nThe eigenfunctions are $e^{ik\\theta}$, $k \\in \\Z$, with eigenvalues $k + a$.\nSo $\\eta_A(s) = \\sum_{k \\ge 0} (k + a)^{-s} - \\sum_{k \\ge 1} (k - a)^{-s} = \\zeta(s, a) - \\zeta(s, 1 - a)$, with $\\zeta(s, a)$ the [[Hurwitz zeta function|G4SW7HJG88]].\nSince $\\zeta(0, a) = \\tfrac12 - a$, the eta invariant is $\\eta_A(0) = 1 - 2a$.\nFor $a = 0$ the nonzero spectrum $\\Z \\setminus \\{0\\}$ is symmetric and $\\eta_A(0) = 0$.\n\nNon-example: $B = d/d\\theta$ on $S^1$ is elliptic but not self-adjoint; its eigenvalues $ik$ are imaginary, so $\\operatorname{sign}(\\lambda)$ has no meaning and $B$ has no eta invariant.\n",
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