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.

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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
van der Corput B-process
historicalPeriod early 20th century
influenced methods in additive combinatorics
modern exponential sum techniques
mathematicalDomain harmonic analysis
number theory
namedAfter J. G. van der Corput
relatedConcept van der Corput lemma in harmonic analysis
relatedTo Hardy–Littlewood circle method
Vinogradov’s method
Weyl differencing
linked to: Weyl’s method

Weyl sums
Weyl’s inequality
linked to: Weyl inequalities

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

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Referenced by (10)

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
Vinogradov's three-primes theorem usesConcept Weyl differencing
linked to: van der Corput method for estimating exponential sums
van der Corput lemma relatedTo Weyl differencing
linked to: van der Corput method for estimating exponential sums
van der Corput method hasVariant van der Corput differencing method
linked to: van der Corput method for estimating exponential sums
van der Corput method relatedTo Vinogradov’s method
linked to: van der Corput method for estimating exponential sums
van der Corput inequality isPartOf van der Corput method for exponential sums
linked to: van der Corput method for estimating exponential sums
van der Corput method for estimating exponential sums relatedTo Weyl sums
linked to: van der Corput method for estimating exponential sums
van der Corput method for estimating exponential sums hasVariant van der Corput A-process
linked to: van der Corput method for estimating exponential sums
van der Corput method for estimating exponential sums hasVariant van der Corput B-process
linked to: van der Corput method for estimating exponential sums
theory of uniform distribution modulo 1 studies Weyl sums
linked to: van der Corput method for estimating exponential sums