GRH

E898459

mathematical conjecture number theory conjecture unproven conjecture

GRH is a major unproven conjecture in number theory asserting that all nontrivial zeros of a broad class of L-functions lie on a critical line, generalizing the classical Riemann hypothesis.

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All labels observed (1)

Label	Occurrences
GRH canonical	1

Statements (45)

Predicate	Object
instanceOf	mathematical conjecture ⓘ number theory conjecture ⓘ unproven conjecture ⓘ
abbreviationOf	Generalized Riemann Hypothesis NERFINISHED ⓘ
appliesTo	Dirichlet L-functions associated to primitive characters NERFINISHED ⓘ a broad class of L-functions beyond the Riemann zeta function ⓘ
asserts	all nontrivial zeros of certain L-functions lie on the critical line Re(s) = 1/2 ⓘ
concerns	Dirichlet L-functions NERFINISHED ⓘ L-functions ⓘ zeros of L-functions ⓘ
field	analytic number theory ⓘ number theory ⓘ
fullName	Generalized Riemann Hypothesis NERFINISHED ⓘ
generalizes	Riemann Hypothesis NERFINISHED ⓘ
hasConsequence	constraints on possible counterexamples to the Riemann Hypothesis ⓘ sharper bounds for class numbers of number fields ⓘ
hasVariant	Extended Riemann Hypothesis NERFINISHED ⓘ Grand Riemann Hypothesis NERFINISHED ⓘ
implies	effective versions of the Chebotarev density theorem ⓘ improved bounds in computational number theory algorithms ⓘ results on distribution of primes in arithmetic progressions ⓘ results on least quadratic nonresidues ⓘ strong bounds on error terms in prime number theorems for arithmetic progressions ⓘ various bounds in algebraic number theory ⓘ
involves	complex analysis ⓘ prime number distribution in arithmetic progressions ⓘ
isPartOf	Millennium Prize Problems context ⓘ
isStrongerThan	Riemann Hypothesis NERFINISHED ⓘ
logicalStatus	independent of current axioms is unknown ⓘ
motivation	generalizing properties of the Riemann zeta function ⓘ understanding distribution of primes ⓘ
openQuestion	whether any nontrivial zero of Dirichlet L-functions lies off the critical line ⓘ
relatedTo	Dedekind zeta functions NERFINISHED ⓘ Dirichlet characters NERFINISHED ⓘ Riemann Hypothesis NERFINISHED ⓘ automorphic L-functions ⓘ critical line Re(s) = 1/2 ⓘ critical strip 0 < Re(s) < 1 ⓘ zeta functions ⓘ
status	open problem ⓘ unproven ⓘ
studiedBy	number theorists ⓘ
usedIn	conditional bounds for algorithms in primality testing and factoring ⓘ conditional results in algebraic number theory ⓘ conditional results in computational complexity ⓘ

Referenced by (1)

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

generalized Riemann hypothesis → alsoKnownAs → GRH ⓘ