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.
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| basedOn | Riemann’s unpublished notes ⓘ |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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.