Siegel’s theorem on zeros of L-functions
E747888
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
All labels observed (4)
How this entity was disambiguated
This entity first appeared as the object of triple T8644722 — 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’s theorem on zeros of L-functions Context triple: [Deuring–Heilbronn phenomenon, relatedTo, Siegel’s theorem on zeros of L-functions]
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A.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
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B.
Euler products for automorphic L-functions
Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
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C.
Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
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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.
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E.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
- 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’s theorem on zeros of L-functions Target entity description: Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
-
A.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
B.
Euler products for automorphic L-functions
Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
-
C.
Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
-
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.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result about Dirichlet L-functions
ⓘ
theorem in analytic number theory ⓘ |
| appliesTo |
Dirichlet L-functions
ⓘ
L-functions associated to Dirichlet characters ⓘ |
| assumes | Dirichlet characters modulo q ⓘ |
| concerns |
nontrivial zeros of L-functions
ⓘ
zeros near 1 of Dirichlet L-functions ⓘ zeros of Dirichlet L-functions ⓘ |
| context |
classical theory of Dirichlet L-series
ⓘ
study of the distribution of primes in residue classes ⓘ |
| describes |
exceptional real zeros of L(s, χ) close to 1
ⓘ
location of real zeros of L(s, χ) with χ real ⓘ |
| feature |
ineffectivity of the implied constants
ⓘ
possible existence of an exceptional real zero (Siegel zero) ⓘ |
| field | analytic number theory ⓘ |
| gives |
bounds on how close nontrivial zeros can approach 1
ⓘ
lower bounds for 1 − β where β is a real zero of a Dirichlet L-function ⓘ |
| hasAlternativeName |
Siegel’s theorem on exceptional zeros
ⓘ
linked to:
Siegel’s theorem on zeros of L-functions
Siegel’s theorem on real zeros of Dirichlet L-functions ⓘ
linked to:
Siegel’s theorem on zeros of L-functions
|
| hasConsequence |
effective bounds for primes in arithmetic progressions up to a possible exceptional modulus
ⓘ
ineffective constants in some prime distribution estimates ⓘ strong error terms in the prime number theorem for arithmetic progressions ⓘ |
| hasProperty | non-effective (ineffective) nature of the constant in the bound ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies |
restrictions on Siegel zeros
ⓘ
strong bounds on exceptional real zeros of Dirichlet L-functions ⓘ zeros of Dirichlet L-functions cannot lie too close to 1 ⓘ |
| influenced |
later work on zero-free regions for L-functions
ⓘ
research on eliminating or controlling Siegel zeros ⓘ |
| involves |
Dirichlet L-function L(s, χ)
ⓘ
linked to:
Dirichlet L-functions
real characters modulo q ⓘ zeros on the real axis close to s = 1 ⓘ |
| isAbout | how close a real zero of L(s, χ) can be to 1 in terms of the modulus q ⓘ |
| mathematicalDomain |
complex analysis
ⓘ
number theory ⓘ |
| namedAfter | Carl Ludwig Siegel ⓘ |
| provedBy | Carl Ludwig Siegel ⓘ |
| provides | ineffective lower bounds for 1 − β in terms of the modulus q ⓘ |
| relatedTo |
Dirichlet’s theorem on arithmetic progressions
ⓘ
Generalized Riemann Hypothesis ⓘ
linked to:
generalized Riemann hypothesis
Landau–Page theorem ⓘ Siegel zero ⓘ prime number theorem for arithmetic progressions ⓘ zero-free regions for L-functions ⓘ |
| usedFor | distribution of primes in arithmetic progressions ⓘ |
| usedIn |
bounding error terms in Chebotarev-type results for abelian extensions
ⓘ
proofs of strong versions of the prime number theorem in arithmetic progressions ⓘ |
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
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Input
Subject: Siegel’s theorem on zeros of L-functions Description of subject: Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions
→
hasAlternativeName
→
Siegel’s theorem on exceptional zeros
ⓘ
linked to: Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions
→
hasAlternativeName
→
Siegel’s theorem on real zeros of Dirichlet L-functions
ⓘ
linked to: Siegel’s theorem on zeros of L-functions