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 |
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
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Instruction
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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)