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  "body": "\n## Definition\n\nLet $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], and for $z \\in \\mathbb{H}$ let $q = e^{2 \\pi i z}$.\nThe *Ramanujan tau function* $\\tau \\colon \\Z_{\\ge 1} \\to \\Z$ is defined by the expansion\n$$q \\prod_{m = 1}^{\\infty} (1 - q^m)^{24} = \\sum_{n = 1}^{\\infty} \\tau(n) q^n.$$\nThe left side is the [[cusp form|A50Y012RBP]] $\\Delta(z)$ of weight $12$.\n\n## Remarks\n\nExample: the coefficient of $q^2$ comes only from $q \\cdot (1 - q)^{24}$, which gives $-24$; so $\\tau(2) = -24$.\nThe function is multiplicative: $\\tau(mn) = \\tau(m)\\tau(n)$ when $\\gcd(m, n) = 1$.\nFor primes $p$ and $r \\ge 1$, $\\tau(p^{r+1}) = \\tau(p)\\tau(p^r) - p^{11} \\tau(p^{r-1})$.\nNon-example: $\\tau$ is not completely multiplicative, since $\\tau(4) = \\tau(2)^2 - 2^{11} = -1472 \\neq 576 = \\tau(2)^2$.\nRamanujan's congruence: $\\tau(n) \\equiv \\sigma_{11}(n) \\pmod{691}$, where $\\sigma_{11}(n) = \\sum_{d \\mid n} d^{11}$.\nFor $n = 2$: $\\sigma_{11}(2) - \\tau(2) = 2049 + 24 = 2073 = 3 \\cdot 691$.\nWhether $\\tau(n) \\neq 0$ for all $n$ is an open question, known as Lehmer's conjecture.\n",
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