van der Corput method for estimating exponential sums

E776136

analytic number theory method technique for bounding exponential sums

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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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 ⓘ

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Johannes G. van der Corput → notableConcept → van der Corput method for estimating exponential sums ⓘ