Riemann zeta function
E47609
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.
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 L-functions → |
| growthProperty |
ζ(s) is of order 1 as an entire function of s after removing pole
→
|
| historicalPaper |
Riemann 1859 memoir "On the Number of Primes Less Than a Given Magnitude"
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|
| introducedBy |
Bernhard Riemann
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|
| 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) → |