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  "body": "\n## Definition\n\nLet $k \\ge 4$ be an even integer and let $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]].\nThe *Eisenstein series of weight $k$* is the function $G_k \\colon \\mathbb{H} \\to \\C$ given by\n$$G_k(\\tau) = \\sum_{(m, n) \\in \\Z^2 \\setminus \\set{(0, 0)}} \\frac{1}{(m\\tau + n)^k}.$$\nThe *normalized Eisenstein series* is $E_k = G_k / (2\\zeta(k))$, where $\\zeta$ is the [[Riemann zeta function|1XWBY4N0VT]].\n\n## Remarks\n\nFor $k \\ge 3$ the series converges absolutely and locally uniformly on $\\mathbb{H}$.\nSo $G_k$ is holomorphic, and it is a [[modular form|GWKN5VG78A]] of weight $k$ for $\\mathrm{SL}_2(\\Z)$.\nWith $q = e^{2 \\pi i \\tau}$, $\\sigma_{k-1}(n) = \\sum_{d \\mid n} d^{k-1}$ (a [[divisor function|04J4Z73J5B]]) and $B_k$ the $k$-th [[Bernoulli number|0KC00K7KY4]]:\n$$E_k(\\tau) = 1 - \\frac{2k}{B_k} \\sum_{n \\ge 1} \\sigma_{k-1}(n) q^n.$$\nExample: $B_4 = -1/30$ gives $E_4 = 1 + 240 \\sum_{n \\ge 1} \\sigma_3(n) q^n$, and $B_6 = 1/42$ gives $E_6 = 1 - 504 \\sum_{n \\ge 1} \\sigma_5(n) q^n$.\nNon-example: for odd $k$, the terms for $(m, n)$ and $(-m, -n)$ cancel, so the sum is $0$; this is why $k$ is even.\nFor $k = 2$ the double series does not converge absolutely, and the function $E_2$ obtained by summing in a fixed order is not a modular form.\n",
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