Dirichlet L-functions

E259755

L-function complex analytic function object of analytic number theory

Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.

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All labels observed (7)

Label	Occurrences
Dirichlet L-functions canonical	25
Dirichlet L-function	6
Dirichlet L-function modulo q	2
Dirichlet L-function Euler products	1
Dirichlet L-function L(s, χ)	1
Dirichlet L-functions associated to primitive characters	1
Dirichlet L-functions modulo q	1

How this entity was disambiguated

This entity first appeared as the object of triple T2364373 — 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: Dirichlet L-functions
Context triple: [Riemann hypothesis, relatedTo, Dirichlet L-functions]

A. Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
B. Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
C. Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
D. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
E. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
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: Dirichlet L-functions
Target entity description: Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.

A. Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
B. Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
C. Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
D. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
E. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	L-function ⓘ complex analytic function ⓘ object of analytic number theory ⓘ
basedOn	Dirichlet characters ⓘ
centralIn	Dirichlet’s theorem on arithmetic progressions ⓘ linked to: Dirichlet's theorem on arithmetic progressions
conjecture	Generalized Riemann Hypothesis for Dirichlet L-functions ⓘ linked to: generalized Riemann hypothesis
convergesAbsolutelyOn	half-plane Re(s)>1 ⓘ
dependsOn	modulus q of Dirichlet character ⓘ primitive or imprimitive nature of character χ ⓘ
domain	complex plane ⓘ
encodes	arithmetic information about residue classes modulo q ⓘ
extendsTo	meromorphic function on C ⓘ
generalizes	Riemann zeta function ⓘ
GRHStatement	all nontrivial zeros lie on line Re(s)=1/2 ⓘ
hasCoefficient	Dirichlet character values χ(n) ⓘ
hasComponent	completed L-function Λ(s,χ) ⓘ
hasEulerProduct	L(s,χ)=∏_{p}(1−χ(p)p^{-s})^{-1} for Re(s)>1 ⓘ
hasGeneralization	L-functions attached to Grössencharacters ⓘ
hasSeriesDefinition	L(s,χ)=∑_{n=1}^{∞} χ(n)n^{-s} for Re(s)>1 ⓘ
hasSpecialCase	Dirichlet beta function as L(s,χ) for nontrivial character mod 4 ⓘ Riemann zeta function ζ(s)=L(s,χ₀) for trivial character χ₀ modulo 1 ⓘ linked to: Riemann zeta function
hasSymmetry	functional equation symmetric about Re(s)=1/2 ⓘ
hasTool	explicit formula relating zeros and primes in arithmetic progressions ⓘ
hasZeroType	nontrivial zeros ⓘ trivial zeros ⓘ
implies	infinitely many primes in each admissible arithmetic progression ⓘ
involves	Gamma factors in functional equation ⓘ conductor of Dirichlet character ⓘ parity of Dirichlet character ⓘ
multiplicativityProperty	Dirichlet characters are completely multiplicative ⓘ
namedAfter	Peter Gustav Lejeune Dirichlet ⓘ
nontrivialZerosRegion	critical strip 0<Re(s)<1 ⓘ
nonvanishingProperty	L(1,χ)≠0 for nontrivial Dirichlet characters χ ⓘ
poleProperty	L(s,χ) is entire for nonprincipal characters ⓘ
relatedTo	Artin L-functions ⓘ Hecke L-functions ⓘ linked to: L-functions automorphic L-functions ⓘ
satisfies	analytic continuation except possible simple pole at s=1 for principal characters ⓘ functional equation relating L(s,χ) and L(1−s,χ̄) ⓘ
studiedIn	analytic number theory ⓘ
trivialZerosLocation	negative integers depending on parity of χ ⓘ
usedIn	Chebotarev density theorem ⓘ analytic class number formula for imaginary quadratic fields ⓘ class number formulas for quadratic fields ⓘ distribution of primes in residue classes ⓘ study of primes in arithmetic progressions ⓘ
usedToDefine	Dirichlet density of sets of primes ⓘ
variable	complex variable s ⓘ

How these facts were elicited

Referenced by (37)

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

Riemann hypothesis → relatedTo → Dirichlet L-functions ⓘ

Riemann zeta function → generalization → Dirichlet L-functions ⓘ

Euler product formula for the Riemann zeta function → relatesTo → Dirichlet L-functions ⓘ

Peter Gustav Lejeune Dirichlet → notableWork → Dirichlet L-functions ⓘ

Multiplicative Number Theory → relatedTo → Dirichlet L-functions ⓘ

Legendre symbol → usedIn → Dirichlet L-functions ⓘ

Deuring–Heilbronn phenomenon → involves → Dirichlet L-function ⓘ

linked to: Dirichlet L-functions

Deuring–Heilbronn phenomenon → appliesTo → Dirichlet L-function modulo q ⓘ

linked to: Dirichlet L-functions

E. C. Titchmarsh, The Theory of the Riemann Zeta-Function → relatedConcept → Dirichlet L-functions ⓘ

Selberg class → contains → Dirichlet L-functions ⓘ

generalized Riemann hypothesis → statementAbout → Dirichlet L-functions ⓘ

generalized Riemann hypothesis → appliesTo → Dirichlet L-functions modulo q ⓘ

linked to: Dirichlet L-functions

Dirichlet series → specialCase → Dirichlet L-function ⓘ

linked to: Dirichlet L-functions

Euler products for automorphic L-functions → generalizes → Dirichlet L-function Euler products ⓘ

linked to: Dirichlet L-functions

L-function → hasSpecialCase → Dirichlet L-function ⓘ

subject linked to: L-functions

linked to: Dirichlet L-functions

Dirichlet characters → associatedWith → Dirichlet L-functions ⓘ

Kummer congruences → relatedTo → Dirichlet L-functions ⓘ

Dirichlet → knownFor → Dirichlet L-functions ⓘ

Dirichlet's theorem on arithmetic progressions → usesConcept → Dirichlet L-functions ⓘ

Dirichlet's theorem on arithmetic progressions → relatedTo → Dirichlet L-function ⓘ

linked to: Dirichlet L-functions

Lindelöf hypothesis → relatedTo → Dirichlet L-functions ⓘ

algebraic number theory → fieldOfStudy → Dirichlet L-functions ⓘ

Bombieri–Vinogradov theorem → involves → Dirichlet L-functions ⓘ

Vorlesungen über Zahlentheorie → associatedWith → Dirichlet L-functions ⓘ

subject linked to: Vorlesungen über Zahlentheorie (with Dirichlet)

Mellin transform → usedWith → Dirichlet L-functions ⓘ

subject linked to: Mellin transforms

Multiplicative Number Theory I. Classical Theory → topic → Dirichlet L-functions ⓘ

subject linked to: Multiplicative Number Theory I. Classical Theory (Hugh L. Montgomery, Robert C. Vaughan)

Siegel zero → isZeroOf → Dirichlet L-function ⓘ

linked to: Dirichlet L-functions

Siegel zero → associatedWith → Dirichlet L-function modulo q ⓘ

linked to: Dirichlet L-functions

Siegel’s theorem on zeros of L-functions → appliesTo → Dirichlet L-functions ⓘ

Siegel’s theorem on zeros of L-functions → involves → Dirichlet L-function L(s, χ) ⓘ

linked to: Dirichlet L-functions

Dirichlet density → relatedTo → Dirichlet L-functions ⓘ

Siegel–Walfisz theorem → uses → Dirichlet L-functions ⓘ

GRH → concerns → Dirichlet L-functions ⓘ

GRH → appliesTo → Dirichlet L-functions associated to primitive characters ⓘ

linked to: Dirichlet L-functions

Artin L-functions → generalize → Dirichlet L-functions ⓘ

Artin L-functions → specialCase → Dirichlet L-function ⓘ

linked to: Dirichlet L-functions

Hurwitz zeta function → relatedTo → Dirichlet L-functions ⓘ

All labels observed (7)

How this entity was disambiguated Show

Statements (48)

How these facts were elicited Show

Referenced by (37)

How this entity was disambiguated

How these facts were elicited