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  "body": "\n## Summary\n\nThe Ramanujan tau function $\\tau(n)$ gives the Fourier coefficients of the [[cusp form|A50Y012RBP]] $\\Delta = q \\prod_{n \\ge 1} (1 - q^n)^{24}$ of weight $12$.\nRamanujan studied it in 1916 and conjectured that it is multiplicative, which Mordell proved in 1917.\nIts properties come from the theory of [[Hecke operators|S4SV3QZ8B1]].\nThe bound $\\abs{\\tau(p)} \\le 2 p^{11/2}$ for primes $p$ is the [[Ramanujan–Petersson conjecture|4CDFWST967]], proved by Deligne.\n\n## Explanations\n\n### L1 human\n\nMultiply out the infinite product $q(1 - q)^{24}(1 - q^2)^{24}(1 - q^3)^{24} \\cdots$ and read off the coefficient of $q^n$; that is $\\tau(n)$.\nThe first values are $\\tau(1) = 1$, $\\tau(2) = -24$, $\\tau(3) = 252$, $\\tau(4) = -1472$ and $\\tau(5) = 4830$.\nThese numbers look random, yet $\\tau(6) = \\tau(2)\\tau(3)$, and in general $\\tau(mn) = \\tau(m)\\tau(n)$ when $m$ and $n$ share no factor.\nThey also satisfy the congruence $\\tau(n) \\equiv \\sigma_{11}(n) \\pmod{691}$.\n",
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