
This article is cited in 6 scientific papers (total in 6 papers)
Methods of estimating of incomplete Kloosterman sums
M. A. Korolev^{} ^{} Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Abstract:
This survey contains enlarged version of a minicourse which was read by the author in November 2015 during “Chinese 
Russian workshop of exponential sums and sumsets”. This workshop was organized by professors Chaohua Jia
(Institute of Mathematics, Academia Sinica) and Ke Gong (Henan University) in Academy of Mathematics and System Science, CAS (Beijing). The author is warmly grateful to them for the support and hospitality.
The survey contains the Introduction, three parts and Conclusion. The basic definitions and results concerning the
complete Kloosterman sums are given in the Introduction.
The method of estimating of incomplete Kloosterman sums to moduli equal to the raising power of a fixed prime is described in the first part. This method is based on one idea of A. G. Postnikov which reduces the estimate of such sums to the estimate of
the exponential sums with polynomial by I. M. Vinogradov's mean value theorem.
A. A. Karatsuba's method of estimating of incomplete sums to an arbitrary moduli is described in the second part.
This method is based on a very precise estimate of the number of solutions of one symmetric congruence involving inverse residues to a given modulus. This estimate plays the same role in thie problems under considering as Vinogradov's mean value theorem in the estimating of corresponding exponential sums.
The method of J. Bourgain and M. Z. Garaev is described in the third part. This method is based on very deep “sumproduct estimate” and on the improvement of A. A. Karatsuba's bound for the number of solutions of symmetric congruence.
The Conclusion contains a series of recent results concerning the estimates of short Kloosterman sums.
Bibliography: 57 titles.
Keywords:
inverse residues, incomplete Kloosterman sums, method of Postnikov, method of Karatsuba, method of Bourgain and Garaev, Vinogradov's mean value theorem, sumproduct estimate.
DOI:
https://doi.org/10.22405/22268383201617479109
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UDC:
511.321 Received: 22.04.2016 Accepted:12.12.2016
Citation:
M. A. Korolev, “Methods of estimating of incomplete Kloosterman sums”, Chebyshevskii Sb., 17:4 (2016), 79–109
Citation in format AMSBIB
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\by M.~A.~Korolev
\paper Methods of estimating of incomplete Kloosterman sums
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 4
\pages 79109
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\crossref{https://doi.org/10.22405/22268383201617479109}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2362830}
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This publication is cited in the following articles:

M. A. Korolev, “On Anatolii Alekseevich Karatsuba's works written in the 1990s and 2000s”, Proc. Steklov Inst. Math., 299 (2017), 1–43

M. A. Korolev, “Kloosterman sums with multiplicative coefficients”, Izv. Math., 82:4 (2018), 647–661

M. A. Korolev, “Elementary Proof of an Estimate for Kloosterman Sums with Primes”, Math. Notes, 103:5 (2018), 761–768

M. A. Korolev, “New estimate for a Kloosterman sum with primes for a composite modulus”, Sb. Math., 209:5 (2018), 652–659

M. A. Korolev, “Divisors of a quadratic form with primes”, Proc. Steklov Inst. Math., 303 (2018), 154–170

M. A. Korolev, “Short Kloosterman Sums with Primes”, Math. Notes, 106:1 (2019), 89–97

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