Statements (23)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:alsoKnownAs |
gptkb:Riemann_zeta_function
|
| gptkbp:category |
gptkb:Analytic_number_theory
Special functions |
| gptkbp:definedIn |
ζ(s) = ∑_{n=1}^∞ 1/n^s
|
| gptkbp:domain |
complex numbers s with Re(s) > 1
|
| gptkbp:generalizes |
gptkb:Hurwitz_zeta_function
|
| gptkbp:hasEquation |
ζ(s) = 2^s π^{s-1} sin(πs/2) Γ(1-s) ζ(1-s)
|
| gptkbp:hasEulerProduct |
ζ(s) = ∏_{p prime} (1 - p^{-s})^{-1}
|
| https://www.w3.org/2000/01/rdf-schema#label |
Euler's zeta function
|
| gptkbp:introduced |
gptkb:Leonhard_Euler
|
| gptkbp:meromorphicContinuation |
entire complex plane except s=1
|
| gptkbp:poleAt |
s=1
|
| gptkbp:relatedTo |
gptkb:Prime_number_theorem
gptkb:Bernoulli_numbers |
| gptkbp:residueAt |
s=1, residue=1
|
| gptkbp:specialValueAt |
ζ(2) = π^2/6
ζ(4) = π^4/90 |
| gptkbp:usedIn |
gptkb:Riemann_hypothesis
gptkb:Number_theory Analytic continuation |
| gptkbp:bfsParent |
gptkb:Leonard_Euler
|
| gptkbp:bfsLayer |
7
|