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 | 2 |
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-function modulo q
ⓘ
linked to:
Dirichlet L-functions
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-function
ⓘ
linked to:
Dirichlet L-functions
Dirichlet character ⓘ
linked to:
Dirichlet characters
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 problem ⓘ
linked to:
Siegel zero
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
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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 (2)
Full triples — surface form annotated when it differs from this entity's canonical label.