Math Conjectures

Generalized Riemann Hypothesis for Dirichlet LL-functions

OPENLandmarkConjectureProposed c. 1884 · Standard version

Canonical statement

For every primitive Dirichlet character χ\chi modulo an integer q≥1q\ge1, every zero ρ\rho of the analytically continued Dirichlet function
L(s,χ)=∑n=1∞χ(n)ns(ℜs>1) L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}\qquad(\Re s>1)
satisfying 0<ℜρ<10<\Re\rho<1 has ℜρ=12\Re\rho=\tfrac12.
View source LaTeX
For every primitive Dirichlet character \(\chi\) modulo an integer \(q\ge1\), every zero \(\rho\) of the analytically continued Dirichlet function
\[
  L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}\qquad(\Re s>1)
\] satisfying \(0<\Re\rho<1\) has \(\Re\rho=\tfrac12\).
NT-AN-003:[Davenport2000MNT][IwaniecKowalski2004][Conrey2003RH]
Number theoryAnalysis and functional analysis

Notes

The Generalized Riemann Hypothesis (GRH), in the classical Dirichlet form recorded here, asserts that for every primitive Dirichlet character χ\chi modulo q≥1q\ge1, all zeros of the continued function L(s,χ)L(s,\chi) in the strip 0<ℜs<10<\Re s<1 lie on the line ℜs=12\Re s=\tfrac12. The extension of Riemann's prediction to Dirichlet LL-functions took shape in the late nineteenth century, around 1884, without a uniquely dated formulation; the label GRH is also used for Dedekind, Artin, and automorphic variants not treated in this record.

What is known parallels the case q=1q=1. Classical zero-free regions exclude zeros near ℜs=1\Re s=1, apart from a possible exceptional real zero attached to a quadratic character, and these suffice for the prime number theorem in arithmetic progressions in restricted ranges [Davenport2000MNT]. Assumed as a hypothesis, GRH gives square-root-type error terms for primes in progressions and underlies a large body of conditional results in analytic number theory [IwaniecKowalski2004]; the surrounding conjectural landscape is surveyed by Conrey [Conrey2003RH].

The hypothesis remains open: no proof covers all primitive characters, and even the case q=1q=1, which is the Riemann Hypothesis itself, is unresolved.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.