Hurwitz zeta function
E931273
Dirichlet series
complex analytic function
generalization of Riemann zeta function
meromorphic function
special function
The Hurwitz zeta function is a complex analytic function that generalizes the Riemann zeta function by introducing a shift parameter, playing a key role in analytic number theory and special function theory.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hurwitz zeta function canonical | 5 |
| Hurwitz–Lerch zeta function (as a special case with z = 1) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11534415 — 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: Hurwitz zeta function Context triple: [Adolf Hurwitz, knownFor, Hurwitz zeta 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 eta function
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.
-
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.
Riemann xi function
The Riemann xi function is an entire, symmetrized version of the Riemann zeta function that encodes its nontrivial zeros and plays a central role in the study of the Riemann Hypothesis and related analytic 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: Hurwitz zeta function Target entity description: The Hurwitz zeta function is a complex analytic function that generalizes the Riemann zeta function by introducing a shift parameter, playing a key role in analytic number theory and special function theory.
-
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 eta function
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.
-
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.
Riemann xi function
The Riemann xi function is an entire, symmetrized version of the Riemann zeta function that encodes its nontrivial zeros and plays a central role in the study of the Riemann Hypothesis and related analytic number theory.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Dirichlet series
ⓘ
complex analytic function ⓘ generalization of Riemann zeta function ⓘ meromorphic function ⓘ special function ⓘ |
| analyticContinuationDomain | C × (C \ Z_{≤0}) except s = 1 ⓘ |
| appearsIn |
analytic number theory
ⓘ
quantum field theory ⓘ spectral theory ⓘ statistical mechanics ⓘ |
| coefficientInLaurentExpansion | Stieltjes constants generalized by parameter a ⓘ |
| convergesFor | Re(s) > 1 ⓘ |
| definingConditionOnParameter | Re(a) > 0 ⓘ |
| definition | ζ(s,a) = Σ_{n=0}^{∞} (n + a)^{-s} for Re(s) > 1 and Re(a) > 0 ⓘ |
| differentiationRelation | ∂/∂a ζ(s,a) = -s ζ(s+1,a) ⓘ |
| domainOfParameter | complex numbers ⓘ |
| domainOfVariable | complex numbers ⓘ |
| fieldOfStudy |
complex analysis
ⓘ
number theory ⓘ special function theory ⓘ |
| functionalEquationType | generalized functional equation extending that of Riemann zeta function ⓘ |
| generalizes |
Hurwitz–Lerch zeta function (as a special case with z = 1)
ⓘ
linked to:
Hurwitz zeta function
Riemann zeta function ⓘ |
| growthProperty | of polynomial growth in vertical strips away from s = 1 ⓘ |
| hasAnalyticContinuation | yes ⓘ |
| hasLaurentExpansionAt | s = 1 ⓘ |
| hasSimplePoleAt | s = 1 ⓘ |
| namedAfter | Adolf Hurwitz ⓘ |
| parameter | a ⓘ |
| relatedTo |
Bernoulli numbers
ⓘ
Bernoulli polynomials ⓘ Dirichlet L-functions ⓘ Gamma function ⓘ Riemann zeta function ⓘ polygamma functions ⓘ polylogarithm ⓘ |
| residueAt | (s = 1, residue = 1) ⓘ |
| seriesType | Dirichlet series in (n+a)^{-s} ⓘ |
| specialCase |
ζ(s,1) = Riemann zeta function ζ(s)
ⓘ
ζ(s,1/2) related to Dirichlet L-function L(s,χ_2) ⓘ ζ(s,a) with rational a expressible via Dirichlet L-functions ⓘ |
| symbol | ζ(s,a) ⓘ |
| usedFor |
evaluation of series
ⓘ
regularization of divergent sums ⓘ special values at integers ⓘ study of distribution of arithmetic sequences ⓘ |
| valueAt |
ζ(-n,a) = -B_{n+1}(a)/(n+1) for n ∈ N
ⓘ
ζ(0,a) = 1/2 - a ⓘ |
| variable | 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: Hurwitz zeta function Description of subject: The Hurwitz zeta function is a complex analytic function that generalizes the Riemann zeta function by introducing a shift parameter, playing a key role in analytic number theory and special function theory.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.