Siegel zero
E747887
A Siegel zero is a hypothetical exceptional real zero of certain Dirichlet L-functions that would lie unusually close to 1 and have deep implications for the distribution of prime numbers in arithmetic progressions.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Siegel zero canonical | 2 |
| Landau–Siegel zero | 1 |
| Landau–Siegel zeros | 1 |
| Siegel zero problem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8644707 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Siegel zero Context triple: [Deuring–Heilbronn phenomenon, involves, Siegel zero]
-
A.
Bombieri–Vinogradov theorem
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
-
B.
Lindelöf hypothesis
The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
-
C.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
-
D.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
E.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Siegel zero Target entity description: A Siegel zero is a hypothetical exceptional real zero of certain Dirichlet L-functions that would lie unusually close to 1 and have deep implications for the distribution of prime numbers in arithmetic progressions.
-
A.
Bombieri–Vinogradov theorem
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
-
B.
Lindelöf hypothesis
The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
-
C.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
-
D.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
E.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
concept in analytic number theory
ⓘ
exceptional zero of Dirichlet L-function ⓘ hypothetical real zero ⓘ |
| affects |
Chebotarev density theorem error terms
ⓘ
distribution of primes in residue classes ⓘ error term in prime number theorem for arithmetic progressions ⓘ |
| alsoCalled |
Landau–Siegel zero
ⓘ
linked to:
Siegel zero
|
| associatedWith |
Dirichlet L-function modulo q
ⓘ
linked to:
Dirichlet L-functions
quadratic Dirichlet character ⓘ real primitive Dirichlet character ⓘ |
| constrainedBy |
Siegel’s theorem on L(1,χ)
ⓘ
linked to:
Siegel’s theorem on zeros of L-functions
zero-free regions for Dirichlet L-functions ⓘ |
| dependsOn | modulus q of the Dirichlet character ⓘ |
| discussedIn |
literature on exceptional zeros of L-functions
ⓘ
research on primes in arithmetic progressions ⓘ |
| hasApproximateForm | β with β>1−c/(log q) for some small c>0 ⓘ |
| hasConsequence |
exceptional bias in Chebyshev’s bias
ⓘ
exceptionally good lower bounds for L(1,χ) ⓘ large deviations from expected distribution of primes mod q ⓘ |
| hasImplication |
ineffective constants in some number theoretic estimates
ⓘ
ineffectivity in lower bounds for class numbers of quadratic fields ⓘ |
| hasOpenProblem |
existence of Siegel zeros for any modulus
ⓘ
nonexistence of Siegel zeros for all Dirichlet L-functions ⓘ |
| hasProperty |
real
ⓘ
simple zero ⓘ |
| hasRealPart | very close to 1 ⓘ |
| implies | strong irregularities in distribution of primes in arithmetic progressions ⓘ |
| isExcludedFor | many small moduli q by explicit computations ⓘ |
| isHypothetical | true ⓘ |
| isNear | pole of Riemann zeta function at s=1 ⓘ |
| isPartOf |
study of zeros of L-functions
ⓘ
theory of Dirichlet L-functions ⓘ |
| isZeroOf |
Dirichlet L-function
ⓘ
linked to:
Dirichlet L-functions
L(s,χ) for a real Dirichlet character χ ⓘ |
| liesIn | critical strip of Dirichlet L-function ⓘ |
| liesOn | real axis ⓘ |
| mathematicalField | number theory ⓘ |
| mathematicalSubfield | analytic number theory ⓘ |
| namedAfter | Carl Ludwig Siegel ⓘ |
| relatedTo |
Deuring–Heilbronn phenomenon
ⓘ
Generalized Riemann Hypothesis ⓘ
linked to:
generalized Riemann hypothesis
Landau–Siegel zeros ⓘ
linked to:
Siegel zero
Siegel’s lower bound for class numbers ⓘ class numbers of imaginary quadratic fields ⓘ |
| status |
not known to be impossible
ⓘ
unproven to exist ⓘ |
| studiedIn | analytic theory of L-functions ⓘ |
| wouldCause | exceptional behavior of class numbers ⓘ |
| wouldViolate | standard zero-free region near s=1 for Dirichlet L-functions ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Siegel zero Description of subject: A Siegel zero is a hypothetical exceptional real zero of certain Dirichlet L-functions that would lie unusually close to 1 and have deep implications for the distribution of prime numbers in arithmetic progressions.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Siegel zero
linked to: Siegel zero
linked to: Siegel zero