Riemann–Siegel theta function
E239279
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Riemann–Siegel theta function canonical | 3 |
| Riemann–Siegel phase function | 1 |
| Riemann–Siegel theta function θ(t) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171586 — 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: Riemann–Siegel theta function Context triple: [Riemann–Siegel formula, hasPart, Riemann–Siegel theta function]
-
A.
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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B.
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.
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C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
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E.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
- 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: Riemann–Siegel theta function Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
-
E.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical function
ⓘ
special function ⓘ |
| alternativeName |
Riemann–Siegel theta function
ⓘ
surface form:
Riemann–Siegel phase function
|
| appearsIn |
Riemann–Siegel formula
ⓘ
surface form:
Riemann–Siegel explicit formula
Siegel’s work on the Riemann zeta function ⓘ Titchmarsh’s theory of the Riemann zeta function ⓘ theory of the distribution of zeros of ζ(s) ⓘ |
| approximationQuality | high accuracy for large |t| ⓘ |
| argumentOf | Hardy Z-function ⓘ |
| codomain | real numbers ⓘ |
| constructedFrom | completed Riemann zeta function ξ(s) ⓘ |
| definedUsing | log Γ((1/4)+it/2) ⓘ |
| domain | real numbers ⓘ |
| expresses | phase of the Riemann zeta function on the critical line ⓘ |
| field |
analytic number theory
ⓘ
complex analysis ⓘ |
| hasAsymptoticExpansion | (t/2) log(t/2π) − t/2 − π/8 + O(1/t) ⓘ |
| hasSeriesExpansion | asymptotic series in descending powers of t ⓘ |
| namedAfter |
Bernhard Riemann
ⓘ
Carl Ludwig Siegel ⓘ |
| property |
monotonically increasing for sufficiently large t
ⓘ
real-valued for real argument t ⓘ smooth function of t ⓘ |
| relatedTo |
Gram points
ⓘ
Hardy Z-function ⓘ Riemann hypothesis ⓘ
surface form:
Riemann Hypothesis
Riemann zeta function ⓘ Stirling's approximation ⓘ
surface form:
Stirling’s approximation
argument of ζ(1/2+it) ⓘ functional equation of the Riemann zeta function ⓘ logarithm of the gamma function ⓘ |
| symbol | θ(t) ⓘ |
| usedFor |
Riemann–Siegel formula
ⓘ
asymptotic analysis of the Riemann zeta function ⓘ computation of zeros of the Riemann zeta function ⓘ counting zeros of ζ(s) via Gram points ⓘ defining Gram points on the critical line ⓘ efficient evaluation of ζ(1/2+it) ⓘ high-precision computation of the Riemann zeta function ⓘ locating high zeros of the Riemann zeta function ⓘ study of the Riemann zeta function on the critical line ⓘ transforming ζ(1/2+it) into a real-valued function Z(t) on the critical line ⓘ writing ζ(1/2+it) as Z(t) e^{-iθ(t)} ⓘ |
| usedIn |
computational number theory
ⓘ
verification of zeros of ζ(s) ⓘ |
| usedToExpress | ζ(1/2+it) in terms of a real-valued function ⓘ |
| variable | real variable t ⓘ |
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: Riemann–Siegel theta function Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Riemann–Siegel phase function
this entity surface form:
Riemann–Siegel theta function θ(t)