Selberg–Delange method results
E637300
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Selberg–Delange method | 1 |
| Selberg–Delange method results canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7030754 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Selberg–Delange method results Context triple: [Multiplicative Number Theory, hasClassicResult, Selberg–Delange method results]
-
A.
Selberg sieve
The Selberg sieve is a powerful analytic number theory method developed by Atle Selberg for estimating the size of sets of integers filtered by divisibility conditions, particularly in the study of prime numbers.
-
B.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
-
C.
Dirichlet hyperbola method
The Dirichlet hyperbola method is a technique in analytic number theory used to estimate sums of arithmetic functions by splitting double sums along a hyperbola to obtain asymptotic formulas.
-
D.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
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E.
Erdős–Kac theorem
The Erdős–Kac theorem is a fundamental result in probabilistic number theory stating that the number of distinct prime factors of a typical integer behaves like a normally distributed random variable.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Selberg–Delange method results Target entity description: 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.
-
A.
Selberg sieve
The Selberg sieve is a powerful analytic number theory method developed by Atle Selberg for estimating the size of sets of integers filtered by divisibility conditions, particularly in the study of prime numbers.
-
B.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
-
C.
Dirichlet hyperbola method
The Dirichlet hyperbola method is a technique in analytic number theory used to estimate sums of arithmetic functions by splitting double sums along a hyperbola to obtain asymptotic formulas.
-
D.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
E.
Erdős–Kac theorem
The Erdős–Kac theorem is a fundamental result in probabilistic number theory stating that the number of distinct prime factors of a typical integer behaves like a normally distributed random variable.
- F. None of above. chosen
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
ⓘ
Perron’s formula ⓘ 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
ⓘ
Hubert Delange ⓘ |
| provides | uniform asymptotic estimates in ranges of parameters ⓘ |
| relatedTo |
Halász’s theorem
ⓘ
linked to:
Halász theorem
Tauberian theorems ⓘ Wiener–Ikehara theorem ⓘ
linked to:
Tauberian theorems
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
ⓘ
linked to:
Selberg–Delange method results
complex-analytic techniques ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Selberg–Delange method results Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Selberg–Delange method results