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gptkbp:instanceOf
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gptkb:mathematical_concept
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gptkbp:alsoKnownAs
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gptkb:Riemann_zeta_function
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gptkbp:analyticContinuation
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entire complex plane except s=1
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gptkbp:appearsIn
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complex analysis
mathematical physics
number theory
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gptkbp:centralConjecture
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gptkb:Riemann_Hypothesis
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gptkbp:definedIn
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ζ(s) = ∑_{n=1}^∞ n^{-s}
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gptkbp:domain
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complex numbers s with Re(s) > 1
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gptkbp:EulerProduct
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ζ(s) = ∏_{p prime} (1 - p^{-s})^{-1}
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gptkbp:functionalEquation
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ζ(s) = 2^s π^{s-1} sin(πs/2) Γ(1-s) ζ(1-s)
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gptkbp:generalizes
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gptkb:Dirichlet_L-functions
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gptkbp:hasApplication
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gptkb:probability_theory
gptkb:quantum_physics
gptkb:statistical_mechanics
gptkb:string_theory
chaos theory
cryptography
dynamical systems
random matrix theory
fractal geometry
mathematical statistics
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gptkbp:hasIntegralRepresentation
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ζ(s) = 1/Γ(s) ∫₀^∞ x^{s-1}/(e^x-1) dx
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gptkbp:hasLaurentExpansion
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about s=1
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gptkbp:hasMellinTransform
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ζ(s) = ∫₀^∞ x^{s-1}/(e^x-1) dx
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gptkbp:hasNoZeros
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for Re(s) = 1
for Re(s) > 1
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gptkbp:hasPoleAt
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s=1
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gptkbp:hasSeriesRepresentation
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ζ(s) = 1/(1^s) + 1/(2^s) + 1/(3^s) + ...
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gptkbp:hasSpecialCase
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gptkb:Dirichlet_L-function_with_trivial_character
ζ(-1) = -1/12
ζ(0) = -1/2
ζ(1/2) ≈ -1.4603545
ζ(2) = π^2/6
ζ(4) = π^4/90
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gptkbp:hasSpecialValuesAtEvenPositiveIntegers
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rational multiples of powers of π
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gptkbp:hasSpecialValuesAtNegativeIntegers
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related to Bernoulli numbers
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https://www.w3.org/2000/01/rdf-schema#label
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Riemann zeta function (when χ is trivial)
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gptkbp:nontrivialZeros
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complex numbers with 0 < Re(s) < 1
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gptkbp:relatedTo
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gptkb:Prime_number_theorem
gptkb:Bernoulli_numbers
gptkb:Euler_product
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gptkbp:simplePoleResidue
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1
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gptkbp:studiedBy
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gptkb:Leonhard_Euler
gptkb:Bernhard_Riemann
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gptkbp:usedIn
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analytic number theory
distribution of prime numbers
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gptkbp:zeros
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trivial zeros at negative even integers
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gptkbp:bfsParent
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gptkb:lion
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gptkbp:bfsLayer
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4
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