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  "body": "\n## Definition\n\nLet $q \\ge 1$ be an integer, and let $\\chi$ be a [[Dirichlet character|7XB5VJKDPG]] modulo $q$.\nLet $M$ be an integer and $N \\ge 1$.\nThe *character sum* of $\\chi$ over the interval $(M, M + N]$ is\n$$S_\\chi(M, N) = \\sum_{n = M + 1}^{M + N} \\chi(n).$$\nMore generally, a character sum is a sum $\\sum_{n \\in A} \\chi(f(n))$ over a finite set $A$ of integers, where $f$ is a polynomial with integer coefficients.\n\n## Remarks\n\nExample: let $\\chi$ be the Legendre symbol modulo $5$, so $\\chi(1) = \\chi(4) = 1$, $\\chi(2) = \\chi(3) = -1$ and $\\chi(5) = 0$.\nThen $S_\\chi(0, 3) = 1 - 1 - 1 = -1$ and $S_\\chi(0, 5) = 0$.\nNon-example: the sum $\\sum_{n \\le N} \\mu(n)$ of the [[Möbius function|0FANE7P0DY]] is not a character sum, since $\\mu$ is not a Dirichlet character: $\\mu(4) = 0$ but $\\mu(2)\\mu(2) = 1$, so $\\mu$ is not completely multiplicative.\nOrthogonality gives the complete sum: $S_\\chi(M, q) = 0$ if $\\chi$ is not principal, and $S_\\chi(M, q) = \\varphi(q)$ if $\\chi$ is principal, where $\\varphi$ is [[Euler's totient function|0DAMHCT17R]].\nThus $\\abs{S_\\chi(M, N)} \\le \\varphi(q)$ for every nonprincipal $\\chi$.\nThe Pólya–Vinogradov inequality improves this to $\\abs{S_\\chi(M, N)} \\le C \\sqrt{q} \\log q$ for nonprincipal $\\chi$, with an absolute constant $C$.\n",
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