Dirichlet hyperbola method
E466253
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Dirichlet hyperbola method canonical | 2 |
| Dirichlet divisor problem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4746243 — 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: Dirichlet hyperbola method Context triple: [Peter Gustav Lejeune Dirichlet, notableWork, Dirichlet hyperbola method]
-
A.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
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B.
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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C.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
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D.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
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E.
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.
- 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: Dirichlet hyperbola method Target entity description: 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.
-
A.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
B.
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.
-
C.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
D.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
analytic number theory technique
ⓘ
method in number theory ⓘ |
| appearsIn |
expositions of the Dirichlet divisor problem
ⓘ
standard textbooks on analytic number theory ⓘ |
| appliesTo |
Dirichlet convolution of arithmetic functions
ⓘ
multiplicative arithmetic functions ⓘ summatory divisor function ⓘ summatory function of the Euler totient function φ(n) ⓘ summatory function of the Möbius function μ(n) ⓘ summatory function of the divisor function d(n) ⓘ |
| basedOn | hyperbola splitting of the summation domain ⓘ |
| contrastedWith |
Tauberian theorems
ⓘ
complex analytic methods using Dirichlet series ⓘ |
| coreIdea |
balance ranges of summation to minimize error terms
ⓘ
rewrite a sum of a Dirichlet convolution as a double sum ⓘ split the double sum along the hyperbola mn = x ⓘ use symmetry of the region mn ≤ x in the (m,n)-plane ⓘ |
| field | analytic number theory ⓘ |
| gives | main term plus error term for summatory functions ⓘ |
| hasAdvantage |
often gives elementary proofs without complex analysis
ⓘ
provides explicit control over error terms in many cases ⓘ |
| mathematicalDomain |
analysis
ⓘ
number theory ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet ⓘ |
| relatedTo |
Dirichlet convolution
ⓘ
Euler totient function φ(n) ⓘ
linked to:
Euler’s totient function φ(n)
Möbius function μ(n) ⓘ average order of arithmetic functions ⓘ divisor function d(n) ⓘ summatory functions in number theory ⓘ |
| requires |
control of error terms in truncated sums
ⓘ
estimates for partial sums of arithmetic functions ⓘ |
| typicalProblem |
estimating the average order of arithmetic functions
ⓘ
estimating the sum of d(n) up to x ⓘ evaluating sums of convolutions f*g(n) ⓘ |
| typicalStep |
choose a parameter y with 1 ≤ y ≤ x and split sums at y
ⓘ
rewrite ∑_{n≤x} (f*g)(n) as ∑_{mn≤x} f(m)g(n) ⓘ separate the region mn ≤ x into m ≤ y and m > y parts ⓘ |
| usedFor |
deriving asymptotic formulas for summatory functions
ⓘ
estimating sums of arithmetic functions ⓘ splitting double sums into more tractable parts ⓘ transforming double sums into single sums plus error terms ⓘ |
| usedIn |
elementary proofs in analytic number theory
ⓘ
estimates for divisor problems ⓘ estimates for lattice point counting problems related to mn ≤ x ⓘ proofs of average order results ⓘ |
| yields |
asymptotic formula for the divisor summatory function ∑_{n≤x} d(n)
ⓘ
asymptotic formula for ∑_{n≤x} μ(n) under suitable hypotheses ⓘ asymptotic formula for ∑_{n≤x} τ(n) ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Dirichlet hyperbola method Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.