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  "body": "\n## Definition\n\nFor real $\\alpha$ write $e(\\alpha) = e^{2\\pi i \\alpha}$.\nLet $\\mathcal{S}$ be a finite set of positive integers, and let $s \\ge 1$ and $N \\ge 1$ be integers.\nLet $r(N)$ be the number of $s$-tuples $(x_1, \\ldots, x_s) \\in \\mathcal{S}^s$ with $x_1 + \\cdots + x_s = N$.\nLet $f(\\alpha) = \\sum_{x \\in \\mathcal{S}} e(x\\alpha)$, an [[exponential sum|97PBMQCPPM]].\nThe *circle method* consists of the following steps.\n\n1. Write $r(N) = \\int_0^1 f(\\alpha)^s\\, e(-N\\alpha)\\, d\\alpha$.\n2. Choose a parameter $Q$ and split the circle $[0, 1)$ into *major arcs* $\\mathfrak{M}$ and *minor arcs* $\\mathfrak{m} = [0, 1) \\setminus \\mathfrak{M}$. The major arcs are the points near a rational $a/q$ with $q \\le Q$ and $\\gcd(a, q) = 1$.\n3. On $\\mathfrak{M}$, approximate $f$ by explicit functions to get a main term, usually the product of a singular series and a singular integral.\n4. On $\\mathfrak{m}$, bound $\\abs{f}$ uniformly, for example by Weyl's inequality or Vinogradov's estimates, to show that $\\int_{\\mathfrak{m}}$ is smaller than the main term.\n\n## Remarks\n\nStep 1 holds because $\\int_0^1 e(m\\alpha)\\, d\\alpha$ equals $1$ for $m = 0$ and $0$ for every other integer $m$; expanding $f(\\alpha)^s$ gives one term $e((x_1 + \\cdots + x_s - N)\\alpha)$ for each tuple.\nExample: for $\\mathcal{S} = \\set{1, 2}$, $s = 2$ and $N = 3$, $f(\\alpha)^2 e(-3\\alpha) = e(-\\alpha) + 2 + e(\\alpha)$, whose integral is $2$, matching the tuples $(1, 2)$ and $(2, 1)$.\nNon-example: the interval of integration must be a full period; $\\int_0^{1/2} e(\\alpha)\\, d\\alpha = i/\\pi \\ne 0$, so integrating over $[0, 1/2)$ does not isolate the solutions.\nFor [[Waring's problem|7VWGA444K3]], $\\mathcal{S}$ is the set of $k$-th powers up to $N$; for [[Vinogradov's theorem|C01Z9QRGZY]], one uses primes weighted by $\\log p$.\nHardy and Ramanujan worked with power series $\\sum a_n z^n$ and integrals over a circle $\\abs{z} = \\rho \u003c 1$, which is the origin of the name.\n",
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