Selberg–Delange method results

E637300

asymptotic formula result in analytic number theory theorem in multiplicative number theory

Selberg–Delange method results are asymptotic formulas in analytic number theory that precisely describe the average order and distribution of multiplicative arithmetic functions using complex-analytic techniques.

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Statements (46)

Predicate	Object
instanceOf	asymptotic formula ⓘ result in analytic number theory ⓘ theorem in multiplicative number theory ⓘ
appliesTo	arithmetic functions with Euler product generating series ⓘ completely multiplicative functions ⓘ multiplicative functions ⓘ multiplicative functions with controlled local behavior at primes ⓘ multiplicative functions with polynomial growth on prime powers ⓘ
assumes	Euler product representation of generating Dirichlet series ⓘ control of local factors at primes ⓘ
basedOn	Dirichlet series NERFINISHED ⓘ Perron’s formula NERFINISHED ⓘ analytic properties of Dirichlet series ⓘ contour integration ⓘ saddle-point method ⓘ
concerns	behavior of arithmetic functions up to x as x → ∞ ⓘ
describes	average order of multiplicative arithmetic functions ⓘ distribution of multiplicative arithmetic functions ⓘ
field	analytic number theory ⓘ multiplicative number theory ⓘ
generalizes	classical average order results for multiplicative functions ⓘ
gives	error term estimates for summatory functions ⓘ main term and lower-order terms in asymptotic formulas ⓘ precise asymptotic expansions for summatory functions ⓘ
involves	Mellin transform techniques ⓘ decomposition of Dirichlet series into main and regular parts ⓘ expansion around the dominant pole of a Dirichlet series ⓘ
namedAfter	Atle Selberg NERFINISHED ⓘ Hubert Delange NERFINISHED ⓘ
provides	uniform asymptotic estimates in ranges of parameters ⓘ
relatedTo	Halász’s theorem NERFINISHED ⓘ Tauberian theorems NERFINISHED ⓘ Wiener–Ikehara theorem NERFINISHED ⓘ mean values of multiplicative functions ⓘ
requires	analytic continuation of Dirichlet series ⓘ bounds on Dirichlet series in vertical strips ⓘ control of singularities near s = 1 ⓘ
studies	partial sums of multiplicative functions ⓘ summatory functions of arithmetic functions ⓘ
usedFor	average order of divisor functions ⓘ average order of generalized divisor functions ⓘ average order of multiplicative functions defined by Euler products ⓘ distribution of values of multiplicative functions ⓘ refined asymptotics beyond leading term ⓘ
uses	Selberg–Delange method NERFINISHED ⓘ complex-analytic techniques ⓘ

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Multiplicative Number Theory → hasClassicResult → Selberg–Delange method results ⓘ