grand Riemann hypothesis

E898461

mathematical conjecture unproven hypothesis in number theory

The grand Riemann hypothesis is a far-reaching conjecture in number theory asserting that all nontrivial zeros of all automorphic L-functions lie on a critical line in the complex plane, generalizing the classical and generalized Riemann hypotheses.

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All labels observed (1)

Label	Occurrences
grand Riemann hypothesis canonical	1

Statements (48)

Predicate	Object
instanceOf	mathematical conjecture ⓘ unproven hypothesis in number theory ⓘ
asserts	all nontrivial zeros of all automorphic L-functions lie on a critical line ⓘ
concerns	distribution of zeros in the complex plane ⓘ nontrivial zeros of automorphic L-functions ⓘ
criticalLine	Re(s)=1/2 ⓘ
criticalStrip	0 < Re(s) < 1 ⓘ
difficulty	considered extremely difficult ⓘ
field	L-functions NERFINISHED ⓘ analytic number theory ⓘ automorphic forms ⓘ number theory ⓘ
generalizes	Riemann hypothesis NERFINISHED ⓘ generalized Riemann hypothesis NERFINISHED ⓘ
hasConsequence	refined estimates for distribution of primes in arithmetic progressions ⓘ results on distribution of automorphic spectra ⓘ results on distribution of eigenvalues of Hecke operators ⓘ strong bounds on error terms in prime counting functions ⓘ
implies	Riemann hypothesis NERFINISHED ⓘ generalized Riemann hypothesis ⓘ
importance	central problem in modern number theory ⓘ
involves	algebraic number theory ⓘ complex analysis ⓘ harmonic analysis on groups ⓘ representation theory ⓘ
isStrongerThan	extended Riemann hypothesis NERFINISHED ⓘ generalized Riemann hypothesis NERFINISHED ⓘ
motivatedBy	properties of Dirichlet L-functions ⓘ properties of automorphic L-functions ⓘ properties of the Riemann zeta function ⓘ
nontrivialZeroDefinition	zeros in the critical strip excluding trivial zeros from functional equations ⓘ
openAsOf	2024 ⓘ
relatedTo	Langlands program NERFINISHED ⓘ Selberg class NERFINISHED ⓘ automorphic forms ⓘ prime number distribution ⓘ random matrix theory ⓘ spectral theory of automorphic forms ⓘ
status	open problem ⓘ
subjectOf	research in modern number theory ⓘ
typeOf	zero-free region conjecture ⓘ
usesConcept	Dirichlet series NERFINISHED ⓘ Euler product NERFINISHED ⓘ automorphic L-function NERFINISHED ⓘ automorphic representation ⓘ critical line ⓘ functional equation of L-functions ⓘ
zeroLocationClaim	Re(s)=1/2 for all nontrivial zeros of automorphic L-functions ⓘ

Referenced by (1)

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

generalized Riemann hypothesis → relatedTo → grand Riemann hypothesis ⓘ