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  "body": "\n## Definition\n\nLet $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], and for $\\tau \\in \\mathbb{H}$ let $q = e^{2 \\pi i \\tau}$, so that $\\abs{q} \u003c 1$.\nThe *Dedekind eta function* is the function $\\eta \\colon \\mathbb{H} \\to \\C$ given by\n$$\\eta(\\tau) = e^{\\pi i \\tau / 12} \\prod_{n = 1}^{\\infty} (1 - q^n).$$\n\n## Remarks\n\nThe product converges absolutely and locally uniformly, since $\\sum_n \\abs{q}^n \u003c \\infty$.\nSo $\\eta$ is [[holomorphic|0NAHXR1FRH]] on $\\mathbb{H}$, and it has no zeros there.\nIt satisfies $\\eta(\\tau + 1) = e^{\\pi i / 12} \\eta(\\tau)$ and $\\eta(-1/\\tau) = \\sqrt{-i\\tau}\\, \\eta(\\tau)$, with the principal branch of the square root.\nExample: $\\eta(\\tau)^{24} = q \\prod_{n \\ge 1} (1 - q^n)^{24} = \\Delta(\\tau)$, the [[cusp form|A50Y012RBP]] of weight $12$.\nNon-example: $\\eta$ itself is not invariant under $\\tau \\mapsto \\tau + 1$, because of the factor $e^{\\pi i / 12}$, so it is not a modular form for $\\mathrm{SL}_2(\\Z)$.\nBy the [[pentagonal number theorem|PMYDHW00KF]], $\\prod_{n \\ge 1} (1 - q^n) = \\sum_{m \\in \\Z} (-1)^m q^{m(3m - 1)/2}$.\n",
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