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  "body": "\n## Statement\n\nLet $q \\ge 1$ and $a$ be integers, and let $x \\ge 2$ be real.\nLet\n$$\\psi(x; q, a) = \\sum_{\\substack{n \\le x \\\\ n \\equiv a \\pmod q}} \\Lambda(n),$$\nwhere $\\Lambda$ is the [[von Mangoldt function|0NK8JTE1NN]], and let $\\varphi$ be [[Euler's totient function|0DAMHCT17R]].\nFor every real number $N \u003e 0$ there is a constant $C_N \u003e 0$ such that\n$$\\psi(x; q, a) = \\frac{x}{\\varphi(q)} + O\\!\\left(x\\, e^{-C_N \\sqrt{\\log x}}\\right)$$\nfor all $x \\ge 2$, all $q \\le (\\log x)^N$ and all $a$ coprime to $q$.\nThe constant $C_N$ and the implied constant in the [[big O|0TGV80E6A4]] depend only on $N$.\n\n## Remarks\n\nThe proof follows the proof of the [[prime number theorem|0EHBY5Y84N]] for each [[Dirichlet L-function|9D2Z543EJQ]] $L(s, \\chi)$ with $\\chi$ modulo $q$.\nThe classical zero-free region handles all zeros except a possible [[Siegel zero|5VBZBXSFDQ]].\nSiegel's theorem says that for every $\\varepsilon \u003e 0$ such a real zero $\\beta$ satisfies $\\beta \u003c 1 - c(\\varepsilon) q^{-\\varepsilon}$, and for $q \\le (\\log x)^N$ this is enough to make its contribution small.\nSiegel's theorem is ineffective, so $C_N$ cannot be computed from the proof.\nFor example, with $N = 2$ and $q = 4$, the theorem gives $\\psi(x; 4, 1) = x/2 + O(x e^{-c\\sqrt{\\log x}})$, a strong form of [[Dirichlet's theorem on primes in arithmetic progressions|0GNVN3CA2P]] for primes $p \\equiv 1 \\pmod 4$.\nThe range $q \\le (\\log x)^N$ is very short; for larger $q$ one has only average results such as the [[Bombieri–Vinogradov theorem|B3JMY54GJF]].\n\n## Explanations\n\n### L1 human\n\nPrimes are spread evenly over the residue classes modulo $q$ that they can reach.\nThis theorem gives a precise error bound for that fact, valid as long as the modulus $q$ is small compared to $x$, at most a power of $\\log x$.\nIt is the prime number theorem for arithmetic progressions, uniform in small moduli.\n\n### L2 human\n\nThe Siegel–Walfisz theorem is the standard input for the \"small moduli\" part of many arguments in analytic number theory.\nIn proofs of the [[Bombieri–Vinogradov theorem|B3JMY54GJF]] and in the circle method for [[Vinogradov's theorem|C01Z9QRGZY]], the major arcs with small denominators are handled by it.\nIts constants are ineffective because of the possible [[Siegel zero|5VBZBXSFDQ]]; this is why results that rely on it, such as Vinogradov's theorem in its first form, give no explicit bound.\nUnder the [[generalized Riemann hypothesis|DNKA4WBKDS]], the error improves to $O(x^{1/2} (\\log x)^2)$ for all $q \\le x$.\n",
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