generalized Riemann hypothesis

E259754

conjecture in number theory mathematical conjecture unproven hypothesis

The generalized Riemann hypothesis is a major unproven conjecture in number theory asserting that the nontrivial zeros of all Dirichlet L-functions lie on a critical line in the complex plane, extending the classical Riemann hypothesis.

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All labels observed (3)

Label	Occurrences
generalized Riemann hypothesis canonical	2
Generalized Riemann Hypothesis for Dirichlet L-functions	1
extended Riemann hypothesis	1

Statements (48)

Predicate	Object
instanceOf	conjecture in number theory ⓘ mathematical conjecture ⓘ unproven hypothesis ⓘ
alsoKnownAs	GRH ⓘ
appliesTo	Dirichlet L-functions ⓘ surface form: Dirichlet L-functions modulo q Dirichlet characters ⓘ primitive Dirichlet characters ⓘ
asserts	all nontrivial zeros of Dirichlet L-functions lie on the critical line Re(s) = 1/2 ⓘ
concerns	critical strip 0 < Re(s) < 1 ⓘ location of zeros of L-functions ⓘ
criticalLine	Re(s) = 1/2 ⓘ
domain	complex plane ⓘ
extends	Riemann hypothesis ⓘ
field	number theory ⓘ
generalizes	Riemann zeta function case ⓘ
hasConsequence	sharper bounds in many arithmetic counting functions ⓘ zero-free regions for Dirichlet L-functions off the critical line ⓘ
implies	bounds on least prime in an arithmetic progression ⓘ bounds on least quadratic nonresidue modulo a prime ⓘ improved error terms in the prime number theorem for arithmetic progressions ⓘ results on distribution of primes in residue classes ⓘ strong bounds on prime numbers in arithmetic progressions ⓘ
importance	central conjecture in analytic number theory ⓘ major unsolved problem in mathematics ⓘ
involves	Dirichlet characters ⓘ surface form: Dirichlet characters modulo q Euler products ⓘ analytic continuation of L-functions ⓘ
namedAfter	Bernhard Riemann ⓘ
nontrivialZerosLieOn	critical line Re(s) = 1/2 ⓘ
openAsOf	2024 ⓘ
relatedTo	Lindelöf hypothesis ⓘ Riemann hypothesis ⓘ generalized Riemann hypothesis self-linksurface differs ⓘ surface form: extended Riemann hypothesis generalized Lindelöf hypothesis ⓘ grand Riemann hypothesis ⓘ
statementAbout	Dirichlet L-functions ⓘ zeros of L-functions ⓘ
status	open problem ⓘ unproven ⓘ
subfield	L-function theory ⓘ algebraic number theory ⓘ analytic number theory ⓘ
typeOfZero	nontrivial zeros ⓘ
usedIn	algorithmic number theory ⓘ complexity theory conditional results ⓘ computational number theory ⓘ cryptography conditional analyses ⓘ
yearFormulatedApprox	19th century ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann hypothesis → hasGeneralization → generalized Riemann hypothesis ⓘ

Deuring–Heilbronn phenomenon → relatedTo → generalized Riemann hypothesis ⓘ

generalized Riemann hypothesis → relatedTo → generalized Riemann hypothesis self-linksurface differs ⓘ

this entity surface form: extended Riemann hypothesis

Dirichlet L-functions → conjecture → generalized Riemann hypothesis ⓘ

this entity surface form: Generalized Riemann Hypothesis for Dirichlet L-functions