Dirichlet eta function
E466247
The Dirichlet eta function is an alternating Dirichlet series closely related to the Riemann zeta function and used in analytic number theory, particularly for studying series convergence and analytic continuation.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Dirichlet eta function canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T4746237 — 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 eta function Context triple: [Peter Gustav Lejeune Dirichlet, notableWork, Dirichlet eta function]
-
A.
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.
-
B.
Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
-
C.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
-
D.
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.
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E.
Ramanujan tau function
The Ramanujan tau function is a multiplicative arithmetic function arising from the Fourier coefficients of a modular discriminant form, central to the study of modular forms and number theory.
- 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 eta function Target entity description: The Dirichlet eta function is an alternating Dirichlet series closely related to the Riemann zeta function and used in analytic number theory, particularly for studying series convergence and analytic continuation.
-
A.
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.
-
B.
Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
-
C.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
-
D.
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.
-
E.
Ramanujan tau function
The Ramanujan tau function is a multiplicative arithmetic function arising from the Fourier coefficients of a modular discriminant form, central to the study of modular forms and number theory.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Dirichlet series
ⓘ
mathematical function ⓘ special function ⓘ |
| alsoKnownAs | alternating zeta function NERFINISHED ⓘ |
| analyticContinuation | entire function ⓘ |
| category | L-function ⓘ |
| classification | alternating Dirichlet L-series with principal character modulo 2 ⓘ |
| codomain | complex numbers ⓘ |
| convergenceProperty |
absolutely convergent for Re(s) > 1
ⓘ
conditionally convergent for 0 < Re(s) ≤ 1 ⓘ converges for Re(s) > 0 ⓘ |
| definition | η(s) = ∑_{n=1}^{∞} (-1)^{n-1} / n^s ⓘ |
| definitionDomain | Re(s) > 0 ⓘ |
| domain | complex plane ⓘ |
| EulerTransformation | admits Euler summation acceleration ⓘ |
| expansionAround | admits Taylor expansion around s = 0 ⓘ |
| field |
analytic number theory
ⓘ
complex analysis ⓘ |
| functionalEquationRelation | related to functional equation of Riemann zeta function ⓘ |
| growthProperty | of finite order as an entire function ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet NERFINISHED ⓘ |
| property |
Dirichlet series with real coefficients
ⓘ
alternating sign coefficients ⓘ entire extension obtained via relation to ζ(s) ⓘ |
| relatedConcept |
Dirichlet beta function
NERFINISHED
ⓘ
Dirichlet lambda function NERFINISHED ⓘ Hurwitz zeta function NERFINISHED ⓘ |
| relatedTo | Riemann zeta function NERFINISHED ⓘ |
| relationToBernoulliNumbers | values at negative integers expressible via Bernoulli numbers ⓘ |
| relationToPolylogarithm | η(s) = (1 - 2^{1-s}) Li_s(1) ⓘ |
| relationToZeta | η(s) = (1 - 2^{1-s}) ζ(s) ⓘ |
| seriesRepresentation | η(s) = 1 - 2^{-s} + 3^{-s} - 4^{-s} + ⋯ ⓘ |
| seriesType | alternating series ⓘ |
| singularityStructure | no poles in the complex plane ⓘ |
| symbol | η(s) ⓘ |
| usedFor |
analytic continuation of the Riemann zeta function
ⓘ
regularization of divergent series ⓘ studying convergence of Dirichlet series ⓘ |
| usedIn |
study of the Riemann hypothesis
ⓘ
summation of Grandi-type series ⓘ |
| valueAt |
η(0) = 1/2
ⓘ
η(1) = ln(2) ⓘ η(2) = π^2 / 12 ⓘ η(−1) = 1/4 ⓘ η(−2n) = 0 for positive integer n ⓘ |
| zeroType |
has trivial zeros at negative even integers
ⓘ
nontrivial zeros correspond to nontrivial zeros of ζ(s) ⓘ |
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: Dirichlet eta function Description of subject: The Dirichlet eta function is an alternating Dirichlet series closely related to the Riemann zeta function and used in analytic number theory, particularly for studying series convergence and analytic continuation.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
eta