Math 2.0

Number theory Analytic number theory Conjecture

Generalized Riemann hypothesis

Builds on Dirichlet characterDirichlet L-functionZero of a holomorphic functionDedekind zeta functionAlgebraic number fieldSiegel zero

Statement

Let χ\chi be a Dirichlet character and let L(s,χ)L(s, \chi) be its Dirichlet L-function, continued to all of ℂ\C. Then every zero ss of L(s,χ)L(s, \chi) with 0<Re⁡s<10 < \operatorname{Re} s < 1 satisfies Re⁡s=1/2\operatorname{Re} s = 1/2.

Remarks

The conjecture is open. For the principal character modulo 11, L(s,χ)L(s, \chi) is the Riemann zeta function, so the conjecture contains the Riemann hypothesis. The extended Riemann hypothesis makes the same claim for the Dedekind zeta function of every number field. The conjecture implies that there is no Siegel zero. It also implies ψ(x;q,a)=x/φ(q)+O(x1/2(log⁡x)2)\psi(x; q, a) = x/\varphi(q) + O(x^{1/2} (\log x)^2) for all q≤xq \le x and all aa coprime to qq, with an absolute implied constant. Under the conjecture, the Miller test becomes a deterministic primality test: it suffices to test the bases a<2(ln⁡n)2a < 2(\ln n)^2, a bound of Bach. The Bombieri–Vinogradov theorem proves an average form of these consequences without any hypothesis.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptDedekind zeta function, concept, no evidence levelDedekind zeta functionconceptDirichlet L-function, concept, no evidence levelDirichlet L-functionconceptDirichlet character, concept, no evidence levelDirichlet characterconceptSiegel zero, concept, no evidence levelSiegel zeroconceptZero of a holomorphic function, concept, no evidence levelZero of a holomorphic…conceptGeneralized Riemann hypothesis, conjecture, no evidence levelGeneralized Riemann h…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number field concept
  2. Dedekind zeta function concept
  3. Dirichlet L-function concept
  4. Dirichlet character concept
  5. Siegel zero concept
  6. Zero of a holomorphic function concept
  7. Generalized Riemann hypothesis conjecture

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis
About this page
TagDNKA4WBKDS
URIm2://DNKA4WBKDS
Kindconjecture
WritingW0 stub
MSC11M26, 11M06
StatusThis copy of the site has no build data. Run m2 status DNKA4WBKDS.
AboutDirichlet L-function, Riemann hypothesis
UsesDirichlet character, Dirichlet L-function, Zero of a holomorphic function, Dedekind zeta function, Algebraic number field, Siegel zero
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Changed2 November 2026

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