Riemann zeta function

E47609
Dirichlet series L-function complex-valued function meromorphic function special function

The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (4)

Label Occurrences
Riemann zeta function canonical 29
Riemann zeta function ζ(s) 3
Riemann zeta function zeros 1

Statements (51)

Predicate Object
instanceOf Dirichlet series ⓘ
L-function ⓘ
complex-valued function ⓘ
meromorphic function ⓘ
special function ⓘ
analyticContinuation extends meromorphically to C except s = 1 ⓘ
approximateFunctionalEquation ζ(s) expressed as finite sums plus error term ⓘ
BernoulliNumberRelation ζ(2n) = (-1)^{n+1} (B_{2n} (2π)^{2n}) / (2 (2n)!) ⓘ
completedZetaDefinition ξ(s) = ½ s(s-1) π^{-s/2} Γ(s/2) ζ(s) ⓘ
completedZetaProperty ξ(s) is entire ⓘ
completedZetaSymmetry ξ(s) = ξ(1-s) ⓘ
connectedTo prime number theorem ⓘ
criticalLine Re(s) = 1/2 ⓘ
definedOn complex plane ⓘ
DirichletSeriesConvergenceRegion Re(s) > 1 ⓘ
DirichletSeriesDefinition ζ(s) = Σ_{n=1}^{∞} 1/n^s ⓘ
EulerProduct ζ(s) = ∏_{p prime} (1 - p^{-s})^{-1} ⓘ
EulerProductConvergenceRegion Re(s) > 1 ⓘ
field analytic number theory ⓘ
complex analysis ⓘ
functionalEquation ζ(s) = 2^s π^{s-1} sin(πs/2) Γ(1-s) ζ(1-s) ⓘ
generalization Dedekind zeta functions ⓘ
Dirichlet L-functions ⓘ
Hasse–Weil zeta function ⓘ
surface form: Hasse–Weil L-functions
growthProperty ζ(s) is of order 1 as an entire function of s after removing pole ⓘ
historicalPaper Über die Anzahl der Primzahlen unter einer gegebenen Grösse ⓘ
surface form: Riemann 1859 memoir "On the Number of Primes Less Than a Given Magnitude"
introducedBy Bernhard Riemann ⓘ
momentStudy moments of ζ(1/2+it) studied in random matrix theory ⓘ
nontrivialZerosRegion 0 < Re(s) < 1 ⓘ
pole simple pole at s = 1 ⓘ
primeNumberTheoremRelation nonvanishing of ζ(s) on Re(s) = 1 equivalent to prime number theorem ⓘ
relatedConjecture Riemann hypothesis ⓘ
relatedTo distribution of prime numbers ⓘ
residueAt1 1 ⓘ
RiemannHypothesisStatement all nontrivial zeros lie on Re(s) = 1/2 ⓘ
specialValue ζ(-1) = -1/12 ⓘ
ζ(0) = -1/2 ⓘ
ζ(1/2) is irrational (conjectured, not proved) ⓘ
ζ(2) = π^2/6 ⓘ
ζ(4) = π^4/90 ⓘ
symbol ζ(s) ⓘ
trivialZero s = -2 ⓘ
s = -4 ⓘ
s = -6 ⓘ
trivialZerosLocation negative even integers ⓘ
universalityProperty Voronin universality theorem ⓘ
valuesAtEvenIntegers ζ(2n) expressed using Bernoulli numbers and powers of π ⓘ
variable complex variable s ⓘ
VoroninUniversality translates of ζ(s) approximate wide classes of analytic functions in critical strip ⓘ
zeroFreeRegion Re(s) > 1 ⓘ
there exists c > 0 such that ζ(s) ≠ 0 for Re(s) ≥ 1 - c/log(|Im(s)|+2) ⓘ

Referenced by (34)

Full triples — surface form annotated when it differs from this entity's canonical label.

Bernhard Riemann → knownFor → Riemann zeta function ⓘ
Riemann–Siegel formula → mainSubject → Riemann zeta function ⓘ
Riemann hypothesis → involves → Riemann zeta function ⓘ
Über die Anzahl der Primzahlen unter einer gegebenen Grösse → topic → Riemann zeta function ⓘ
Über die Anzahl der Primzahlen unter einer gegebenen Grösse → defines → Riemann zeta function ⓘ
Georg → notableWork → Riemann zeta function ⓘ
subject surface form: Georg Friedrich Bernhard Riemann
Friedrich → notableWork → Riemann zeta function ⓘ
subject surface form: Friedrich Bernhard Riemann
Euler product formula for the Riemann zeta function → relatesTo → Riemann zeta function ⓘ
Zeta → symbolFor → Riemann zeta function ⓘ
Bernoulli numbers → relatedTo → Riemann zeta function ⓘ
Multiplicative Number Theory → usesConcept → Riemann zeta function ⓘ
Hasse–Weil zeta function → generalizes → Riemann zeta function ⓘ
de Bruijn–Newman constant → relatedTo → Riemann zeta function ⓘ
Riemann–Siegel theta function → relatedTo → Riemann zeta function ⓘ
Hardy Z-function → dependsOn → Riemann zeta function ⓘ
Hardy Z-function → constructedFrom → Riemann zeta function ⓘ
this entity surface form: Riemann zeta function ζ(s)
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function → mainSubject → Riemann zeta function ⓘ
H. M. Edwards, Riemann’s Zeta Function → mainSubject → Riemann zeta function ⓘ
subject surface form: Riemann’s Zeta Function (book)
A. Ivić, The Riemann Zeta-Function → mainSubject → Riemann zeta function ⓘ
Selberg class → contains → Riemann zeta function ⓘ
Dirichlet L-functions → generalizes → Riemann zeta function ⓘ
Dirichlet L-functions → hasSpecialCase → Riemann zeta function ⓘ
this entity surface form: Riemann zeta function ζ(s)=L(s,χ₀) for trivial character χ₀ modulo 1
random matrix theory → relatedTo → Riemann zeta function ⓘ
this entity surface form: Riemann zeta function zeros
Hilbert–Pólya conjecture → relatedTo → Riemann zeta function ⓘ
Montgomery's pair correlation conjecture → mainSubject → Riemann zeta function ⓘ
prime number theorem → relatedTo → Riemann zeta function ⓘ
Voronin universality theorem → mainObject → Riemann zeta function ⓘ
Voronin universality theorem → relatedTo → Riemann zeta function ⓘ
Dedekind zeta functions → generalizes → Riemann zeta function ⓘ
subject surface form: Dedekind zeta function
Jordan’s totient functions → relatedConcept → Riemann zeta function ⓘ
this entity surface form: Riemann zeta function ζ(s)
Dirichlet series → specialCase → Riemann zeta function ⓘ
Chebyshev functions → relatedTo → Riemann zeta function ⓘ
subject surface form: Chebyshev function ψ(x)
this entity surface form: Riemann zeta function ζ(s)
Mertens’ theorems → mainTopic → Riemann zeta function ⓘ
L-functions → generalizes → Riemann zeta function ⓘ
subject surface form: L-function