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  "body": "\n## Statement\n\nLet $\\chi$ be a [[Dirichlet character|7XB5VJKDPG]] and let $L(s, \\chi)$ be its [[Dirichlet L-function|9D2Z543EJQ]], continued to all of $\\C$.\nThen every [[zero|00GC8ZQP4Q]] $s$ of $L(s, \\chi)$ with $0 \u003c \\operatorname{Re} s \u003c 1$ satisfies $\\operatorname{Re} s = 1/2$.\n\n## Remarks\n\nThe conjecture is open.\nFor the principal character modulo $1$, $L(s, \\chi)$ is the Riemann zeta function, so the conjecture contains the [[Riemann hypothesis|VCGF9YR36X]].\nThe extended Riemann hypothesis makes the same claim for the [[Dedekind zeta function|KXS5DQDFYD]] of every [[number field|MWBDFS2KJ5]].\nThe conjecture implies that there is no [[Siegel zero|5VBZBXSFDQ]].\nIt also implies $\\psi(x; q, a) = x/\\varphi(q) + O(x^{1/2} (\\log x)^2)$ for all $q \\le x$ and all $a$ coprime to $q$, with an absolute implied constant.\nUnder the conjecture, the Miller test becomes a deterministic primality test: it suffices to test the bases $a \u003c 2(\\ln n)^2$, a bound of Bach.\nThe [[Bombieri–Vinogradov theorem|B3JMY54GJF]] proves an average form of these consequences without any hypothesis.\n\n## Explanations\n\n### L1 human\n\nThe Riemann hypothesis says where the zeros of the Riemann zeta function lie: all the nontrivial ones are on one vertical line.\nThe generalized Riemann hypothesis says the same thing for a whole family of similar functions, the Dirichlet L-functions, one for each way of sorting numbers by residue classes.\nIf it is true, primes are spread over residue classes as evenly as one could hope, with a small and explicit error.\nMany results in number theory and in the analysis of algorithms are first proved under this hypothesis.\n",
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      "first_stated": "1884",
      "posed_by": "Adolf Piltz"
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    "name": "Generalized Riemann hypothesis",
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  "link": "https://math2.tamnd.com/node/DNKA4WBKDS/",
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