Riemann–Siegel formula
E48437
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Riemann–Siegel formula canonical | 4 |
| Riemann–Siegel explicit formula | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T373788 — 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 formula Context triple: [Bernhard Riemann, knownFor, Riemann–Siegel formula]
-
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.
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B.
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.
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C.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
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D.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
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E.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
- 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 formula Target entity description: 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.
-
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.
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.
-
C.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
-
D.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
E.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
asymptotic formula
ⓘ
mathematical formula ⓘ result in analytic number theory ⓘ |
| appearsIn | Carl Ludwig Siegel’s 1932 paper on Riemann’s zeta function ⓘ |
| appliesTo |
Riemann zeta function on the critical line
ⓘ
values of ζ(1/2+it) ⓘ |
| approximationType |
asymptotic expansion
ⓘ
saddle-point approximation ⓘ |
| author | Carl Ludwig Siegel ⓘ |
| basedOn | Riemann’s unpublished notes ⓘ |
| comparedTo | O(t^{1+ε}) operations for direct summation ⓘ |
| complexity | O(t^{1/2+ε}) operations for computing ζ(1/2+it) ⓘ |
| domain | large imaginary part t of s=1/2+it ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| generalization |
Riemann–Siegel type formulas for Dirichlet L-functions
ⓘ
Riemann–Siegel type formulas for automorphic L-functions ⓘ |
| gives | approximate functional equation for ζ(1/2+it) ⓘ |
| hasErrorTerm | remainder of size about O(t^{-1/4}) in basic form ⓘ |
| hasPart |
Riemann–Siegel theta function
ⓘ
main sum over n up to N≈√(t/2π) ⓘ oscillatory cosine terms ⓘ remainder term ⓘ |
| hasRefinement | higher-order Riemann–Siegel expansions ⓘ |
| improvesOn | naive Dirichlet series evaluation of ζ(s) ⓘ |
| influenced | development of fast algorithms for L-functions ⓘ |
| involves |
Gamma function
ⓘ
functional equation of the Riemann zeta function ⓘ stationary phase method ⓘ |
| mainSubject | Riemann zeta function ⓘ |
| namedAfter |
Bernhard Riemann
ⓘ
Carl Ludwig Siegel ⓘ |
| relatedTo |
Hardy Z-function
ⓘ
Riemann–Siegel theta function ⓘ |
| standardReference |
A. Ivić, The Riemann Zeta-Function
ⓘ
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function ⓘ H. M. Edwards, Riemann’s Zeta Function ⓘ |
| use |
efficient numerical evaluation of the Riemann zeta function
ⓘ
high-precision computation of ζ(1/2+it) ⓘ study of zeros of the Riemann zeta function ⓘ verification of the Riemann hypothesis for large heights ⓘ |
| usedBy | Hardy’s method for counting zeros on the critical line ⓘ |
| usedIn |
Odlyzko’s large-scale computations of zeta zeros
ⓘ
verification of the first billions of zeros of ζ(s) ⓘ |
| validOn | critical line Re(s)=1/2 ⓘ |
| yearProposed | 1932 ⓘ |
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 formula Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Riemann–Siegel explicit formula