About: Dirichlet series inversion

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In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little known, or at least often forgotten about, way of expressing formulas for arithmetic functions and their summatory functions is to perform an integral transform that inverts the operation of forming the DGF of a sequence. This inversion is analogous to performing an to the generating function of a sequence to express formulas for the series coefficients of a given ordinary generating function.

Attributes	Values
rdfs:label	Dirichlet series inversion (en)
rdfs:comment	In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little known, or at least often forgotten about, way of expressing formulas for arithmetic functions and their summatory functions is to perform an integral transform that inverts the operation of forming the DGF of a sequence. This inversion is analogous to performing an to the generating function of a sequence to express formulas for the series coefficients of a given ordinary generating function. (en)
dcterms:subject	Integer sequences Number theory Analytic number theory
Wikipage page ID	61869767 (xsd:integer)
Wikipage revision ID	1100039513 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime-counting function Mertens function Multiplicative function Mellin transform Prime zeta function Analytic continuation Generating function transformation Generating function Generating functions Antiderivative Arithmetic function Analytic number theory Integer sequences Number theory Dirichlet convolution Dirichlet series Arithmetic functions Analytic number theory Summatory function Perron's formula Integration by parts Real part Dirichlet generating function Dirichlet inverse Abscissa of convergence Perron formula Moebius function Mellin transformation dbr:Inverse_Z-transform
sameAs	Dirichlet series inversion Dirichlet series inversion
dbp:wikiPageUsesTemplate	dbt:Expand_section dbt:Morefootnotes dbt:Reflist dbt:Apostol_IANT
has abstract	In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little known, or at least often forgotten about, way of expressing formulas for arithmetic functions and their summatory functions is to perform an integral transform that inverts the operation of forming the DGF of a sequence. This inversion is analogous to performing an to the generating function of a sequence to express formulas for the series coefficients of a given ordinary generating function. For now, we will use this page as a compendia of "oddities" and oft-forgotten facts about transforming and inverting Dirichlet series, DGFs, and relating the inversion of a DGF of a sequence to the sequence's summatory function. We also use the notation for coefficient extraction usually applied to formal generating functions in some complex variable, by denoting for any positive integer , whenever denotes the DGF (or Dirichlet series) of f which is taken to be absolutely convergent whenever the real part of s is greater than the abscissa of absolute convergence, . The relation of the Mellin transformation of the summatory function of a sequence to the DGF of a sequence provides us with a way of expressing arithmetic functions such that , and the corresponding Dirichlet inverse functions, , by inversion formulas involving the summatory function, defined by In particular, provided that the DGF of some arithmetic function f has an analytic continuation to , we can express the Mellin transform of the summatory function of f by the continued DGF formula as It is often also convenient to express formulas for the summatory functions over the Dirichlet inverse function of f using this construction of a Mellin inversion type problem. (en)
prov:wasDerivedFrom	wikipedia-en:Dirichlet_series_inversion?oldid=1100039513&ns=0
page length (characters) of wiki page	13163 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Dirichlet_series_inversion
is Link from a Wikipage to another Wikipage of	List of things named after Peter Gustav Lejeune Dirichlet
is foaf:primaryTopic of	wikipedia-en:Dirichlet_series_inversion

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