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 Chebyshevskii Sb., 2016, Volume 17, Issue 4, Pages 79–109 (Mi cheb518)

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 mini-course 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 “sum-product 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, sum-product estimate.

 Funding Agency Grant Number Russian Science Foundation 14-11-00433

DOI: https://doi.org/10.22405/2226-8383-2016-17-4-79-109

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Bibliographic databases:

UDC: 511.321
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
\Bibitem{Kor16} \by M.~A.~Korolev \paper Methods of estimating of incomplete Kloosterman sums \jour Chebyshevskii Sb. \yr 2016 \vol 17 \issue 4 \pages 79--109 \mathnet{http://mi.mathnet.ru/cheb518} \crossref{https://doi.org/10.22405/2226-8383-2016-17-4-79-109} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2362830} \elib{https://elibrary.ru/item.asp?id=27708207} 

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This publication is cited in the following articles:
1. M. A. Korolev, “On Anatolii Alekseevich Karatsuba's works written in the 1990s and 2000s”, Proc. Steklov Inst. Math., 299 (2017), 1–43
2. M. A. Korolev, “Kloosterman sums with multiplicative coefficients”, Izv. Math., 82:4 (2018), 647–661
3. M. A. Korolev, “Elementary Proof of an Estimate for Kloosterman Sums with Primes”, Math. Notes, 103:5 (2018), 761–768
4. M. A. Korolev, “New estimate for a Kloosterman sum with primes for a composite modulus”, Sb. Math., 209:5 (2018), 652–659
5. M. A. Korolev, “Divisors of a quadratic form with primes”, Proc. Steklov Inst. Math., 303 (2018), 154–170
6. M. A. Korolev, “Short Kloosterman Sums with Primes”, Math. Notes, 106:1 (2019), 89–97
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