Geodesic
On non-linear Kloosterman sum
Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 140-147.

Voir la notice de l'article provenant de la source Math-Net.Ru

Exponential sums of a special type — so-called Kloosterman sums — play key role in the series of number-theoretic problems concerning the distribution of inverse residues in the residual rings of given modulo $q$. At the same time, in many cases, the estimates of such sums are based on A. Weil's bound of so-called complete Kloosterman sum of prime modulo. This bound allows one to estimate Kloosterman sums of length $N\ge q^{0.5+\varepsilon}$ for any fixed $\varepsilon>0$ with power-saving factor. Weil's bound was proved originally by methods of algebraic geometry. Later, S. A. Stepanov gave an elementary proof of this bound, but this proof was also complete enough. The aim of this paper is to give an elementary proof of Kloosterman sum of length $N\ge q^{0.5+\varepsilon}$, which also leads to power-saving factor. This proof is based on the trick of “additive shift” of the variable of summation which is widely used in different problems of number theory. Bibliography: 15 titles.
Keywords: inverse residues, Kloosterman sums, Weil's bound. 
@article{CHEB_2016_17_1_a10,
     author = {M. A. Korolev},
     title = {On non-linear {Kloosterman} sum},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {140--147},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2016_17_1_a10/}
}
TY  - JOUR
AU  - M. A. Korolev
TI  - On non-linear Kloosterman sum
JO  - Čebyševskij sbornik
PY  - 2016
SP  - 140
EP  - 147
VL  - 17
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2016_17_1_a10/
LA  - ru
ID  - CHEB_2016_17_1_a10
ER  - 
%0 Journal Article
%A M. A. Korolev
%T On non-linear Kloosterman sum
%J Čebyševskij sbornik
%D 2016
%P 140-147
%V 17
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2016_17_1_a10/
%G ru
%F CHEB_2016_17_1_a10
M. A. Korolev. On non-linear Kloosterman sum. Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 140-147. http://geodesic.mathdoc.fr/item/CHEB_2016_17_1_a10/

[1] Vinogradov I. M., Elements of number theory, Dover Publications, 2003 | MR | MR

[2] Salie H., “Über die Kloostermanschen Summen $S(u, v; q)$”, Math. Z., 34 (1931), 91–109 | DOI | MR

[3] Estermann T., “On Kloosterman's sum”, Mathematika, 8:1 (1961), 83–86 | DOI | MR | Zbl

[4] Weil A., “On some exponential sums”, Proc. Nat. Acad. Sci. USA, 34 (1948), 204–207 | DOI | MR | Zbl

[5] Stepanov S. A., “An estimation of Kloosterman sums”, Mathematics of the USSR-Izvestiya, 5:2 (1971), 319–336 | DOI | MR | Zbl

[6] Bourgain J., “More on the sum-product phenomenon in prime fields and its applications”, Int. J. Number Theory, 1 (2005), 1–32 | DOI | MR | Zbl

[7] Baker R. C., “Kloosterman sums with prime variable”, Acta Arith., 152:4 (2012), 351–372 | DOI | MR

[8] Vinogradov I. M., “The method of trigonometrical sums in the theory of numbers”, Tr. Mat. Inst. Steklova, 23, 1947 (Russian) | MR | Zbl

[9] Burgess D. A., “On character sums and primitive roots”, Proc. London Math. Soc., 12:3 (1962), 179–192 | DOI | MR | Zbl

[10] Karatsuba A. A., “Trigonometric sums of a special type and their applications”, Izv. Akad. Nauk SSSR Ser. Mat., 28:1 (1964), 237–248 (Russian) | Zbl

[11] Karatsuba A. A., “Estimates of character sums”, Izv. Akad. Nauk SSSR Ser. Mat., 34:1 (1970), 20–30 (Russian)

[12] Karatsuba A. A., “Sums of characters with prime numbers”, Izv. Akad. Nauk SSSR Ser. Mat., 34:2 (1970), 299–321 (Russian)

[13] Karatsuba A. A., “Character sums with weights”, Izv. Math., 64:2 (2000), 249–263 | DOI | DOI | MR | Zbl

[14] Bourgain J., Garaev M. Z., “Sumsets of reciprocals in prime fields and multilinear Kloosterman sums”, Izv. Math., 78:4 (2014), 656–707 | DOI | DOI | MR | Zbl

[15] Fouvry E., Michel P., “Sur certaines sommes d'exponentielles sur les nombres premiers”, Ann. scient. Éc. Norm. Sup., 31:1 (1998), 93–130 | MR | Zbl