Geodesic
Estimation of Kloosterman Sums with Primes and Its Application
Matematičeskie zametki, Tome 88 (2010) no. 3, pp. 365-373.

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

Suppose that $p$ is a large prime. In this paper, we prove that, for any natural number $N$ the following estimate holds: $$ \max_{(a,p)=1}\biggl|\sum_{q\le N}e^{2\pi iaq^*/p}\biggr|\le(N^{15/16}+N^{2/3}p^{1/4})p^{o(1)}, $$ where $q$ is a prime and $q^*$ is the least natural number satisfying the congruence $qq^*\equiv1\,(\operatorname{mod}p)$. This estimate implies the following statement: if $p>N>p^{16/17+\varepsilon}$, where $\varepsilon>0$, and if we have $\lambda\not\equiv0\,(\operatorname{mod}p)$, then the number $J$ of solutions of the congruence $$ q_1(q_2+q_3)\equiv\lambda\quad(\operatorname{mod}p) $$ for the primes $q_1,q_2,q_3\le N$ can be expressed as $$ J=\frac{\pi(N)^3}p(1+O(p^{-\delta})),\qquad \delta=\delta(\varepsilon)>0. $$ This statement improves a recent result of Friedlander, Kurlberg, and Shparlinski in which the condition $p>N>p^{38/39+\varepsilon}$ was required.
Keywords: Kloosterman sum, Cauchy–Bunyakovskii inequality, Dirichlet's principle, Vinogradov sieve, Dirichlet $L$-function, trigonometric sum, Manholdt function. 
@article{MZM_2010_88_3_a4,
     author = {M. Z. Garaev},
     title = {Estimation of {Kloosterman} {Sums} with {Primes} and {Its} {Application}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {365--373},
     publisher = {mathdoc},
     volume = {88},
     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_3_a4/}
}
TY  - JOUR
AU  - M. Z. Garaev
TI  - Estimation of Kloosterman Sums with Primes and Its Application
JO  - Matematičeskie zametki
PY  - 2010
SP  - 365
EP  - 373
VL  - 88
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2010_88_3_a4/
LA  - ru
ID  - MZM_2010_88_3_a4
ER  - 
%0 Journal Article
%A M. Z. Garaev
%T Estimation of Kloosterman Sums with Primes and Its Application
%J Matematičeskie zametki
%D 2010
%P 365-373
%V 88
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2010_88_3_a4/
%G ru
%F MZM_2010_88_3_a4
M. Z. Garaev. Estimation of Kloosterman Sums with Primes and Its Application. Matematičeskie zametki, Tome 88 (2010) no. 3, pp. 365-373. http://geodesic.mathdoc.fr/item/MZM_2010_88_3_a4/

[1] A. A. Karatsuba, “Summy drobnykh dolei spetsialnogo vida funktsii”, Dokl. RAN, 349:3 (1996), 302 | MR | Zbl

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

[3] E. Fouvry, P. Michel, “Sur certaines sommes d'exponentielles sur les nombres premiers”, Ann. Sci. École Norm. Sup. (4), 31:1 (1998), 93–130 | MR | Zbl

[4] J. B. Friedlander, P. Kurlberg, I. E. Shparlinski, “Products in residue classes”, Math. Res. Lett., 15:6 (2008), 1133–1147 | MR | Zbl

[5] P. Erdős, A. M. Odlyzko, A. Sárközy, “On the residues of products of prime numbers”, Period. Math. Hungar., 18:3 (1987), 229–239 | DOI | MR | Zbl

[6] A. A. Karatsuba, “Drobnye doli spetsialnogo vida funktsii”, Izv. RAN. Ser. matem., 59:4 (1995), 61–80 | MR | Zbl

[7] A. A. Karatsuba, “Analogi summ Kloostermana”, Izv. RAN. Ser. matem., 59:5 (1995), 93–102 | MR | Zbl

[8] M. A. Korolëv, “Nepolnye summy Kloostermana i ikh prilozheniya”, Izv. RAN. Ser. matem., 64:6 (2000), 41–64 | MR | Zbl

[9] I. M. Vinogradov, Osnovy teorii chisel, Nauka, M., 1965 | MR | Zbl

[10] D. R. Heath-Brown, “Almost-primes in arithmetic progressions and short intervals”, Math. Proc. Cambridge Philos. Soc., 83:3 (1978), 357–375 | DOI | MR | Zbl

[11] H. Davenport, Multiplicative Number Theory, Grad. Texts in Math., 74, Springer-Verlag, New York, NY, 2000 | MR | Zbl