van der Corput method for estimating exponential sums
E776136
The van der Corput method for estimating exponential sums is a classical analytic number theory technique that provides bounds for oscillatory sums by exploiting differencing and smoothness properties of the phase function.
All labels observed (1)
| Label | Occurrences |
|---|---|
| van der Corput method for estimating exponential sums canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9070968 — 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: van der Corput method for estimating exponential sums Context triple: [Johannes G. van der Corput, notableConcept, van der Corput method for estimating exponential sums]
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A.
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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B.
Selberg–Delange method results
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.
-
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.
Bombieri–Vinogradov theorem
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
-
E.
Linnik’s theorem on the least prime in an arithmetic progression
Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
- 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: van der Corput method for estimating exponential sums Target entity description: The van der Corput method for estimating exponential sums is a classical analytic number theory technique that provides bounds for oscillatory sums by exploiting differencing and smoothness properties of the phase function.
-
A.
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.
-
B.
Selberg–Delange method results
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.
-
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.
Bombieri–Vinogradov theorem
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
-
E.
Linnik’s theorem on the least prime in an arithmetic progression
Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
analytic number theory method
ⓘ
technique for bounding exponential sums ⓘ |
| appearsIn |
classical texts on analytic number theory
ⓘ
monographs on exponential sums ⓘ |
| appliesTo |
exponential sums
ⓘ
oscillatory sums ⓘ |
| assumes | non-degeneracy conditions on the phase ⓘ |
| basedOn | van der Corput differencing ⓘ |
| canYield | sub-square-root cancellation in favorable cases ⓘ |
| classification | classical method in analytic number theory ⓘ |
| coreIdea |
gain powers of cancellation via differencing
ⓘ
replace original sum by sums of finite differences ⓘ |
| field | analytic number theory ⓘ |
| goal |
exploit cancellation in oscillatory sums
ⓘ
obtain upper bounds for exponential sums ⓘ |
| hasVariant |
van der Corput A-process
NERFINISHED
ⓘ
van der Corput B-process NERFINISHED ⓘ |
| historicalPeriod | early 20th century ⓘ |
| influenced |
methods in additive combinatorics
ⓘ
modern exponential sum techniques ⓘ |
| mathematicalDomain |
harmonic analysis
ⓘ
number theory ⓘ |
| namedAfter | J. G. van der Corput NERFINISHED ⓘ |
| relatedConcept | van der Corput lemma in harmonic analysis ⓘ |
| relatedTo |
Hardy–Littlewood circle method
NERFINISHED
ⓘ
Vinogradov’s method NERFINISHED ⓘ Weyl differencing NERFINISHED ⓘ Weyl sums NERFINISHED ⓘ Weyl’s inequality NERFINISHED ⓘ stationary phase method ⓘ |
| requires |
control of derivatives of the phase
ⓘ
smooth phase function ⓘ |
| toolFor |
bounding error terms in asymptotic formulas
ⓘ
estimating exponential integrals via discretization ⓘ proving equidistribution of polynomial sequences modulo 1 ⓘ |
| typicalApplication |
bounding polynomial exponential sums
ⓘ
bounding trigonometric sums ⓘ estimates for Weyl sums with polynomial phases ⓘ |
| usedIn |
bounds for exponential sums over integers
ⓘ
bounds for exponential sums over primes ⓘ discrepancy theory ⓘ distribution of prime numbers ⓘ equidistribution problems ⓘ |
| uses |
differencing
ⓘ
smoothness properties of the phase function ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: van der Corput method for estimating exponential sums Description of subject: The van der Corput method for estimating exponential sums is a classical analytic number theory technique that provides bounds for oscillatory sums by exploiting differencing and smoothness properties of the phase function.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
Johannes G. van der Corput
→
notableConcept
→
van der Corput method for estimating exponential sums
ⓘ