Statements (48)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
mathematical function
|
| gptkbp:defines |
complex numbers
|
| gptkbp:evaluates |
s = 0 gives -1/2
s = 2 gives π^2/6 s = 3 gives approximately 1.2020569 s = 4 gives π^4/90 |
| gptkbp:hasFunction |
ζ(s)_=_2^s_π^(s-1)_sin(πs/2)_Γ(1-s)_ζ(1-s)
|
| gptkbp:hasSpecialty |
ζ(2n) = (-1)^(n+1) B_(2n) (2π)^(2n) / (2(2n)!)
|
| gptkbp:hasVariants |
ζ(s)_=_Σ_(1/n^s)_for_n=1_to_∞
|
| https://www.w3.org/2000/01/rdf-schema#label |
Riemann zeta function
|
| gptkbp:isActiveIn |
statistical mechanics
|
| gptkbp:isAvenueFor |
signal processing
|
| gptkbp:isCitedBy |
analytic continuation
Euler product formula |
| gptkbp:isConnectedTo |
gptkb:Riemann_Hypothesis
gptkb:Fermat's_Last_Theorem modular forms L-functions analytic number theory Harmonic series |
| gptkbp:isEvaluatedBy |
s = -1 gives -1/12
s = 1/2 gives critical line |
| gptkbp:isIntegratedWith |
gptkb:Bernoulli_numbers
Fourier series |
| gptkbp:isInvolvedIn |
random matrix theory
analytic continuation of Dirichlet series quantum_physics |
| gptkbp:isLocatedIn |
integral representation
sum of reciprocals of prime powers |
| gptkbp:isRelatedTo |
Dirichlet series
generalized zeta functions zeta regularization zeta function of a number field Euler's_formula Mertens'_third_theorem |
| gptkbp:isStudiedIn |
mathematical analysis
complex analysis |
| gptkbp:isSubjectTo |
s = 1/2
|
| gptkbp:isUsedBy |
distribution of prime numbers
|
| gptkbp:isUsedIn |
cryptography
mathematical physics number theory quantum_field_theory |
| gptkbp:nonPatentCitation |
s ≠ 1
|
| gptkbp:numberBuilt |
non-trivial zeros
|
| gptkbp:relatedTo |
prime numbers
|
| gptkbp:symbolizes |
ζ(s)
|
| gptkbp:technologyDomain |
s > 1
|