Riemann xi function
E824087
The Riemann xi function is an entire, symmetrized version of the Riemann zeta function that encodes its nontrivial zeros and plays a central role in the study of the Riemann Hypothesis and related analytic number theory.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
complex-valued function
ⓘ
entire function ⓘ function in analytic number theory ⓘ special function ⓘ |
| alsoKnownAs |
Riemann Xi function
NERFINISHED
ⓘ
Riemann ξ-function NERFINISHED ⓘ |
| analyticity | entire on the complex plane ⓘ |
| appearsIn | Riemann’s 1859 memoir on the number of primes less than a given magnitude NERFINISHED ⓘ |
| category | L-function-related object ⓘ |
| centralConjecture | Riemann Hypothesis is equivalent to all zeros lying on Re(s) = 1/2 ⓘ |
| codomain | complex numbers ⓘ |
| criticalLine | Re(s) = 1/2 ⓘ |
| definedBy | ξ(s) = \tfrac{1}{2} s(s-1) π^{-s/2} Γ(s/2) ζ(s) ⓘ |
| dependsOn |
Gamma function
NERFINISHED
ⓘ
Riemann zeta function NERFINISHED ⓘ complex variable s ⓘ |
| domain | complex plane ⓘ |
| encodesZerosOf | Riemann zeta function NERFINISHED ⓘ |
| field | analytic number theory ⓘ |
| functionalEquation | ξ(s) = ξ(1-s) ⓘ |
| growthType | entire function of finite order ⓘ |
| hasHadamardProduct | ξ(s) = ξ(0) ∏_ρ (1 - s/ρ) where ρ runs over zeros ⓘ |
| introducedBy | Bernhard Riemann NERFINISHED ⓘ |
| invariantUnder |
complex conjugation combined with reflection across Re(s) = 1/2
ⓘ
s ↦ 1-s ⓘ |
| normalization |
constructed to remove pole of ζ(s) at s = 1
ⓘ
constructed to satisfy a simple functional equation ⓘ |
| order | order 1 entire function ⓘ |
| realOn |
real values for real s
ⓘ
real values on the critical line s = 1/2 + it ⓘ |
| relatedTo |
Riemann–Siegel theta function
NERFINISHED
ⓘ
Riemann–von Mangoldt explicit formula NERFINISHED ⓘ completed zeta function ⓘ |
| symbol | ξ(s) NERFINISHED ⓘ |
| symmetry |
even with respect to s = 1/2
ⓘ
ξ(1/2 + z) = ξ(1/2 - z) ⓘ |
| usedFor |
Hadamard product representations related to ζ(s)
ⓘ
explicit formulas in prime number theory ⓘ formulation of the Riemann Hypothesis ⓘ study of distribution of primes ⓘ |
| usedIn |
random matrix theory analogies for zeta zeros
ⓘ
spectral interpretations of the Riemann Hypothesis ⓘ |
| valueAt |
ξ(0) = 1/2
ⓘ
ξ(1) = 1/2 ⓘ |
| zeroCountingFunction | related to the Riemann–von Mangoldt formula for N(T) ⓘ |
| zerosCorrespondTo | nontrivial zeros of the Riemann zeta function ⓘ |
| zeroSet | same nontrivial zeros as ζ(s) ⓘ |
| zeroSymmetry |
zeros symmetric with respect to the critical line Re(s) = 1/2
ⓘ
zeros symmetric with respect to the real axis ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.