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  "body": "\n## Summary\n\nThe Eisenstein series $G_k$ of even weight $k \\ge 4$ is the sum of $(m\\tau + n)^{-k}$ over all pairs of integers $(m, n) \\neq (0, 0)$.\nIt is the simplest [[modular form|GWKN5VG78A]] of weight $k$ for the [[modular group|VEKV1EM77B]] that is not a [[cusp form|A50Y012RBP]].\nIts $q$-expansion involves the [[Riemann zeta function|1XWBY4N0VT]] and the divisor sums $\\sigma_{k-1}(n)$.\nThe series $E_4$ and $E_6$ generate the ring of all modular forms for $\\mathrm{SL}_2(\\Z)$.\n\n## Explanations\n\n### L1 human\n\nTake the lattice of points $m\\tau + n$ in the complex plane and add up the $k$-th powers of their reciprocals.\nChanging the basis of the lattice by a matrix of $\\mathrm{SL}_2(\\Z)$ only reorders the sum, and this gives the modular symmetry.\nThe result has a neat expansion: $E_4 = 1 + 240 \\sum_{n \\ge 1} \\sigma_3(n) q^n$, where $\\sigma_3(n)$ is the sum of the cubes of the divisors of $n$.\nComparing such expansions proves surprising identities between divisor sums.\n",
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