Dirichlet density
E790518
Dirichlet density is a notion of density for subsets of prime numbers defined via Dirichlet series, used to measure how frequently such primes occur in analytic number theory.
Statements (30)
| Predicate | Object |
|---|---|
| instanceOf |
concept in analytic number theory
ⓘ
density for subsets of prime numbers ⓘ notion of density ⓘ |
| appliesTo |
Chebotarev sets of primes
NERFINISHED
ⓘ
sets of primes defined by congruence conditions ⓘ |
| basedOn | Dirichlet series NERFINISHED ⓘ |
| comparedWith |
analytic density
ⓘ
natural density ⓘ |
| contrastsWith | asymptotic density defined by counting function π_A(x) ⓘ |
| definedOn | subsets of prime numbers ⓘ |
| dependsOn | behavior of Dirichlet series near s = 1 ⓘ |
| domain | set of prime numbers ⓘ |
| field | analytic number theory ⓘ |
| generalizationOf | natural density for many arithmetic sets of primes ⓘ |
| hasAlternativeName | Dirichlet analytic density NERFINISHED ⓘ |
| hasKeyIdea | encode a set of primes in a Dirichlet series and study its singularity at s = 1 ⓘ |
| hasProperty |
defined via limiting behavior of Dirichlet series
ⓘ
invariant under finite modification of a set of primes ⓘ may exist when natural density does not ⓘ |
| mathematicalObjectType | real number between 0 and 1 for many natural sets of primes ⓘ |
| namedAfter | Johann Peter Gustav Lejeune Dirichlet NERFINISHED ⓘ |
| relatedTo |
Chebotarev density theorem
NERFINISHED
ⓘ
Dirichlet L-functions NERFINISHED ⓘ analytic continuation of Dirichlet series ⓘ prime number theorem for arithmetic progressions ⓘ |
| usedFor |
measuring frequency of subsets of prime numbers
ⓘ
measuring naturalness of sets of primes ⓘ studying distribution of primes in arithmetic progressions ⓘ |
| usedIn |
formulation of density theorems in algebraic number theory
ⓘ
proofs and statements about distribution of primes in residue classes ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.