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  "body": "\n## Statement\n\nLet $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], and let $\\vartheta(z, \\tau) = \\sum_{n \\in \\Z} \\exp(\\pi i n^2\\tau + 2\\pi i n z)$ be the Jacobi [[theta function|2F5CYQJVAF]] for $z \\in \\C$ and $\\tau \\in \\mathbb{H}$.\nThen for all $z \\in \\C$ and $\\tau \\in \\mathbb{H}$:\n\n1. $\\vartheta(z, \\tau + 1) = \\vartheta(z + \\tfrac12, \\tau)$;\n2. $$\\vartheta\\!\\left(\\frac{z}{\\tau}, -\\frac{1}{\\tau}\\right) = \\sqrt{-i\\tau}\\;\\exp\\!\\left(\\frac{\\pi i z^2}{\\tau}\\right)\\vartheta(z, \\tau),$$\nwhere $\\sqrt{-i\\tau}$ is the principal branch, which is positive when $\\tau$ is on the positive imaginary axis.\n\n## Remarks\n\nPart 1 holds term by term, because $e^{\\pi i n^2} = (-1)^n = e^{2\\pi i n/2}$.\nProof idea for part 2: for fixed $\\tau$, apply the [[Poisson summation formula|GARGJYSGFG]] to the Gaussian $x \\mapsto \\exp(\\pi i x^2\\tau + 2\\pi i x z)$, whose Fourier transform is again a Gaussian, with $-1/\\tau$ in place of $\\tau$.\nBoth sides are holomorphic in $\\tau$, so it is enough to check $\\tau = it$ with $t \u003e 0$ and continue analytically.\nExample: at $z = 0$ and $\\tau = it$ the formula reads $\\sum_{n} e^{-\\pi n^2/t} = \\sqrt{t}\\sum_n e^{-\\pi n^2 t}$, the classical identity behind the functional equation of the Riemann zeta function.\nThe hypothesis $\\tau \\in \\mathbb{H}$ is needed: for real $\\tau$ the series does not converge.\n\n## Explanations\n\n### L1 human\n\nThe theta function is a sum of Gaussian terms.\nReplacing $\\tau$ by $-1/\\tau$ turns narrow Gaussians into wide ones, and the formula says the sum changes only by a simple explicit factor.\nTogether with the shift $\\tau \\mapsto \\tau + 1$ this makes theta a modular object.\n\n### L2 human\n\nThe maps $\\tau \\mapsto \\tau + 1$ and $\\tau \\mapsto -1/\\tau$ generate the [[modular group|VEKV1EM77B]], so the formula describes how $\\vartheta$ behaves under it.\nThe theta constant $\\vartheta(0, \\tau)$ is invariant under $\\tau \\mapsto \\tau + 2$ and picks up the factor $\\sqrt{-i\\tau}$ under $\\tau \\mapsto -1/\\tau$.\nSo it is a [[modular form|GWKN5VG78A]] of weight $1/2$ for the subgroup generated by these two maps.\nRaising to the fourth power and using part 2 gives the transformation law used in counting sums of four squares.\nIn the integral $\\int_0^\\infty (\\vartheta(0, it) - 1)\\,t^{s/2 - 1}\\,dt$, the formula of the example yields Riemann's proof of the functional equation of $\\zeta(s)$.\n",
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