Deuring–Heilbronn phenomenon
E204744
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Deuring–Heilbronn phenomenon canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1822611 — 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: Deuring–Heilbronn phenomenon Context triple: [Max Deuring, notableConcept, Deuring–Heilbronn phenomenon]
-
A.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
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B.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
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C.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
D.
Ulam spiral
The Ulam spiral is a graphical arrangement of the positive integers in a spiral pattern that reveals striking diagonal alignments of prime numbers, suggesting unexpected structure in their distribution.
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E.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
- 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: Deuring–Heilbronn phenomenon Target entity description: The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
A.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
-
B.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
C.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
D.
Ulam spiral
The Ulam spiral is a graphical arrangement of the positive integers in a spiral pattern that reveals striking diagonal alignments of prime numbers, suggesting unexpected structure in their distribution.
-
E.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
phenomenon in analytic number theory
ⓘ
result in analytic number theory ⓘ |
| appliesTo |
Dirichlet L-functions
ⓘ
surface form:
Dirichlet L-function modulo q
L-functions associated to Dirichlet characters ⓘ |
| assumes | existence of an exceptional real zero close to 1 ⓘ |
| concerns |
distribution of primes in arithmetic progressions
ⓘ
location of zeros of Dirichlet L-functions ⓘ |
| context |
analytic techniques in multiplicative number theory
ⓘ
study of exceptional zeros of L-functions ⓘ |
| describes |
effect of Siegel zeros on the distribution of primes in arithmetic progressions
ⓘ
how an exceptional zero forces other zeros away from the real axis ⓘ repulsion of zeros of L-functions ⓘ sharpening of zero-free regions for Dirichlet L-functions ⓘ |
| field | analytic number theory ⓘ |
| formalizes | zero repulsion effect for Dirichlet L-functions ⓘ |
| hasConsequence |
improved zero-free region near s = 1 for non-exceptional characters
ⓘ
repulsion between a Siegel zero and other zeros in the critical strip ⓘ stronger bounds for primes in arithmetic progressions when an exceptional zero exists ⓘ zeros of other Dirichlet L-functions are pushed away from the line Re(s) = 1 ⓘ |
| hasProperty |
gives quantitative zero repulsion conditional on a Siegel zero
ⓘ
non-effective with respect to the size of the exceptional zero ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies | other zeros of related L-functions are bounded away from 1 by an explicit amount ⓘ |
| involves |
Dirichlet L-functions
ⓘ
surface form:
Dirichlet L-function
Dirichlet characters ⓘ
surface form:
Dirichlet character
Siegel zero ⓘ critical strip ⓘ exceptional zero ⓘ real axis ⓘ zero-free region ⓘ zeros of L-functions ⓘ |
| namedAfter |
Hans Heilbronn
ⓘ
Max Deuring ⓘ |
| relatedTo |
Chebotarev density theorem
ⓘ
Linnik’s theorem on the least prime in an arithmetic progression ⓘ Siegel zero ⓘ
surface form:
Siegel zero problem
Siegel’s theorem on zeros of L-functions ⓘ generalized Riemann hypothesis ⓘ zero-free regions for Dirichlet L-functions ⓘ |
| usedIn |
bounds for error terms in prime number theorems for arithmetic progressions
ⓘ
effective versions of results conditional on the nonexistence of Siegel zeros ⓘ |
How these facts were elicited
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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: Deuring–Heilbronn phenomenon Description of subject: The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.