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.
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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| relatedTo |
Dirichlet convolution
NERFINISHED
ⓘ
Euler totient function φ(n) NERFINISHED ⓘ 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) ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.