de Bruijn–Newman constant
E239168
The de Bruijn–Newman constant is a real number arising in the study of the zeros of the Riemann zeta function and related Fourier transforms, central to a refined form of the Riemann Hypothesis.
All labels observed (1)
| Label | Occurrences |
|---|---|
| de Bruijn–Newman constant canonical | 1 |
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical constant
ⓘ
real number ⓘ |
| appearsIn |
study of deformed Riemann xi functions
ⓘ
theory of entire functions of Laguerre–Pólya class ⓘ |
| characterization | threshold between all-real zeros and some non-real zeros for a family of entire functions ⓘ |
| conjecturedRelationToRiemannHypothesis |
Riemann hypothesis
ⓘ
surface form:
Riemann Hypothesis is equivalent to Λ ≤ 0
|
| conjecturedSign | nonnegative ⓘ |
| connectedTo |
Riemann xi function
ⓘ
distribution of zeros of entire functions ⓘ heat-flow deformation of entire functions ⓘ |
| definitionContext | deformation of the Riemann xi function by the heat equation ⓘ |
| field |
Fourier analysis
ⓘ
analytic number theory ⓘ complex analysis ⓘ |
| furtherDevelopedBy | Charles M. Newman ⓘ |
| hasInequalityFormulation | Λ ≥ 0 ⓘ |
| implication |
if Λ > 0 then some deformations of the xi function have non-real zeros
ⓘ
if Λ ≤ 0 then all zeros of the Riemann xi function are real ⓘ |
| introducedBy |
N. G. de Bruijn
ⓘ
surface form:
Nicolaas Govert de Bruijn
|
| knownLowerBound | Λ ≥ 0 ⓘ |
| knownLowerBoundProvedBy |
Brad Rodgers
ⓘ
Terence Tao ⓘ |
| knownLowerBoundYear | 2018 ⓘ |
| mathematicalDomain | theory of zeta and L-functions ⓘ |
| namedAfter |
Charles M. Newman
ⓘ
N. G. de Bruijn ⓘ
surface form:
Nicolaas Govert de Bruijn
|
| openProblem |
determine the exact value of Λ
ⓘ
prove or disprove Λ = 0 ⓘ |
| property | is the infimum of parameters t for which a deformed xi-function has only real zeros ⓘ |
| refinedConjecture | Λ = 0 ⓘ |
| relatedTo |
Fourier transform
ⓘ
Riemann hypothesis ⓘ
surface form:
Riemann Hypothesis
Riemann zeta function ⓘ entire functions with real zeros ⓘ zeros of the Riemann xi function ⓘ |
| role | measures how far the Riemann Hypothesis could fail ⓘ |
| status |
exact value unknown
ⓘ
sign known to be nonnegative ⓘ |
| symbol | Λ ⓘ |
| upperBound | less than 1 ⓘ |
| usedIn | refined formulations of the Riemann Hypothesis ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.