Dirichlet series

E300759

analytic number theory concept complex function mathematical series

A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.

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All labels observed (3)

Label	Occurrences
Dirichlet series canonical	6
Dirichlet generating function	1
Dirichlet generating functions	1

Statements (47)

Predicate	Object
instanceOf	analytic number theory concept ⓘ complex function ⓘ mathematical series ⓘ
canHaveEulerProduct	when coefficients are multiplicative ⓘ
coefficientSequence	(a_n)_{n≥1} ⓘ
convergesIn	right half-plane Re(s) > σ_c ⓘ
domainOfDefinition	subset of the complex plane ⓘ
encodes	information about coefficients a_n ⓘ
field	analytic number theory ⓘ number theory ⓘ
generalizationOf	ordinary generating function with n^{-s} weights ⓘ
hasAbscissaOfAbsoluteConvergence	σ_a ⓘ
hasAbscissaOfConvergence	σ_c ⓘ
hasAbscissaOfUniformConvergence	σ_u ⓘ
hasGeneralForm	∑_{n=1}^{∞} a_n n^{-s} ⓘ
hasProperty	absolute convergence implies uniform convergence on compact subsets of half-planes ⓘ uniqueness of coefficients in domain of convergence ⓘ
hasTypicalRegionOfAnalyticity	half-plane Re(s) > σ_c ⓘ
isStudiedIn	complex analysis ⓘ harmonic analysis ⓘ probabilistic number theory ⓘ
isToolFor	Tauberian theorems ⓘ analytic continuation ⓘ functional equations of L-functions ⓘ prime number theorems in arithmetic progressions ⓘ
isUsedToStudy	L-functions ⓘ arithmetic functions ⓘ distribution of prime numbers ⓘ multiplicative functions ⓘ
mayAdmit	meromorphic continuation beyond initial half-plane ⓘ
namedAfter	Peter Gustav Lejeune Dirichlet ⓘ
productCorrespondsTo	Dirichlet convolution of coefficient sequences ⓘ
relatedTo	Euler product ⓘ Mellin transforms ⓘ surface form: Mellin transform power series ⓘ
satisfiesInequality	σ_c ≤ σ_u ≤ σ_a ⓘ
specialCase	Dirichlet L-functions ⓘ surface form: Dirichlet L-function Dirichlet series self-linksurface differs ⓘ surface form: Dirichlet generating function Hurwitz zeta function ⓘ Riemann zeta function ⓘ
supportsOperation	Dirichlet convolution via product ⓘ termwise addition ⓘ
usedIn	automorphic forms ⓘ proofs of Dirichlet’s theorem on arithmetic progressions ⓘ spectral theory of automorphic Laplacians ⓘ study of modular forms ⓘ
variable	complex variable s ⓘ

Referenced by (8)

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

Euler product formula for the Riemann zeta function → relatesTo → Dirichlet series ⓘ

Divergent Series → topic → Dirichlet series ⓘ

Multiplicative Number Theory → fieldOfStudy → Dirichlet series ⓘ

Multiplicative Number Theory → hasKeyTool → Dirichlet series ⓘ

this entity surface form: Dirichlet generating functions

A. Ivić, The Riemann Zeta-Function → coversTopic → Dirichlet series ⓘ

Dirichlet series → specialCase → Dirichlet series self-linksurface differs ⓘ

this entity surface form: Dirichlet generating function

Euler’s method of rearranging absolutely convergent series → relatedTo → Dirichlet series ⓘ

Mertens’ theorems → relatedTo → Dirichlet series ⓘ