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gptkbp:instanceOf
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gptkb:mathematical_concept
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gptkbp:analyticContinuation
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entire complex plane except s=1
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gptkbp:application
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gptkb:probability_theory
gptkb:statistical_mechanics
number theory
physics
quantum mechanics
random matrix theory
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gptkbp:category
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complex analysis
mathematical analysis
analytic number theory
mathematical constants
special functions
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gptkbp:citation
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gptkb:Riemann,_B._(1859)._"On_the_Number_of_Primes_Less_Than_a_Given_Magnitude"
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gptkbp:criticalLine
|
Re(s) = 1/2
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gptkbp:definedIn
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ζ(s) = ∑_{n=1}^∞ 1/n^s for Re(s) > 1
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gptkbp:domain
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complex numbers
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gptkbp:EulerProduct
|
ζ(s) = ∏_{p prime} (1 - p^{-s})^{-1} for Re(s) > 1
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gptkbp:firstPublished
|
1859
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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:lion
gptkb:Hurwitz_zeta_function
multiple zeta values
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gptkbp:hasConjecture
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gptkb:Riemann_Hypothesis
|
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gptkbp:hasSpecialCase
|
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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|
https://www.w3.org/2000/01/rdf-schema#label
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Riemann zeta function
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gptkbp:namedAfter
|
gptkb:Bernhard_Riemann
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gptkbp:nontrivialZeros
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gptkb:critical_strip_0_<_Re(s)_<_1
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gptkbp:pole
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s=1
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gptkbp:relatedTo
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gptkb:Dedekind_zeta_function
gptkb:Bernoulli_numbers
gptkb:Euler_product_formula
gptkb:Gamma_function
gptkb:Mellin_transform
gptkb:Polylogarithm
Fourier analysis
modular forms
L-functions
prime numbers
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|
gptkbp:simplePoleResidue
|
1
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gptkbp:studiedBy
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gptkb:Leonhard_Euler
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gptkbp:symbol
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ζ(s)
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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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