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  "body": "\n## Summary\n\nAn exponential sum is a finite sum of numbers $e^{2\\pi i f(n)}$ on the unit circle, where $f$ is a real function on the integers.\nThe trivial bound for such a sum is its number of terms, and the aim is to show cancellation, that is, a much smaller bound.\nSuch bounds are central in analytic number theory: they control [[equidistribution modulo one|XM6DH57BWY]], the zeros of zeta functions, and the circle method.\nWeyl sums, [[Gauss sums|EQ6A3ZN48K]] and [[Kloosterman sums|E1SQWQ2PAV]] are important special cases.\n\n## Explanations\n\n### L1 human\n\nEach term $e^{2\\pi i f(n)}$ is a point on the unit circle, like an arrow of length $1$ pointing in the direction $f(n)$.\nAdding the arrows is like a random walk: if the directions vary enough, the arrows cancel and the sum stays small.\nFor example, with $f(n) = n/4$, the arrows point to $i, -1, -i, 1$ and the sum of each block of four is $0$.\nShowing that such cancellation happens is the main problem of the theory.\n",
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