Short Kloosterman Sums with Primes

M. A. Korolev

Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 84-94.

Résumé

A new estimate of the Kloosterman sum with primes modulo a prime number $q$ is obtained, in which the number of summands can be of order $q^{0.5+\varepsilon}$. This estimate refines results obtained earlier by J. Bourgain (2005) and R. Baker (2012).

Détail

Keywords: Kloosterman sum
Mots-clés : primes, inverse residues.

Bibliographie

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

[2] M. A. Korolev, “Korotkie summy Kloostermana s vesami”, Matem. zametki, 88:3 (2010), 415–427 | DOI | MR | Zbl

[3] M. A. Korolev, “Metody otsenok korotkikh summ Kloostermana”, Chebyshevskii sb., 17:4 (2016), 79–109 | DOI | MR | Zbl

[4] M. A. Korolev, “O korotkikh summakh Kloostermana po prostomu modulyu”, Matem. zametki, 100:6 (2016), 838–846 | DOI | MR | Zbl

[5] M. A. Korolev, “Korotkie summy Kloostermana po moschnomu modulyu”, Dokl. AN, 470:5 (2016), 505–507 | DOI | Zbl

[6] M. A. Korolev, “O metode Karatsuby otsenok summ Kloostermana”, Matem. sb., 207:8 (2016), 117–134 | DOI | MR | Zbl

[7] M. A. Korolev, “O nelineinoi summe Kloostermana”, Chebyshevskii sb., 17:1 (2016), 140–147 | MR

[8] M. A. Korolev, “Ob odnom diofantovom neravenstve s obratnymi velichinami”, Analiticheskaya teoriya chisel, Tr. MIAN, 299, MAIK «Nauka/Interperiodika», M., 2017, 144–154 | DOI

[9] M. A. Korolev, “Obobschennaya summa Kloostermana s prostymi chislami”, Analiticheskaya i kombinatornaya teoriya chisel, Tr. MIAN, 296, MAIK «Nauka/Interperiodika», M., 2017, 163–180 | DOI | MR

[10] M. A. Korolev, “Novaya otsenka summy Kloostermana s prostymi chislami po sostavnomu modulyu”, Matem. sb., 209:5 (2018), 54–61 | DOI | Zbl

[11] E. Fouvry, I. E. Shparlinsky, “On a ternary quadratic form over primes”, Acta Arith., 150:3, 285–314 | DOI | MR | Zbl

[12] M. A. Korolev, “O delitelyakh kvadratichnoi formy s prostymi chislami”, Garmonicheskii analiz, teoriya priblizhenii i teoriya chisel, Tr. MIAN, 303, MAIK «Nauka/Interperiodika», M., 2018, 169–185 | DOI

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

[14] M. Z. Garaev, “Otsenka summ Kloostermana s prostymi chislami i primenenie”, Matem. zametki, 88:3 (2010), 365–373 | DOI | MR

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

[16] R. C. Baker, “Kloosterman sums with prime variable”, Acta Arith., 156:4 (2012), 351–372 | DOI | MR | Zbl

[17] Zh. Burgein, M. Z. Garaev, “Summa mnozhestv, obrazovannykh obratnymi elementami v polyakh prostogo poryadka, i polilineinye summy Kloostermana”, Izv. RAN. Ser. matem., 78:4 (2014), 19–72 | DOI | MR | Zbl

[18] A. Ayyad, N. Cochrane, Z. Zheng, “The congruence $x_1x_2\equiv x_3x_4\,(\operatorname{mod}p)$, the equation $x_1x_2=x_3x_4$, and mean value of character sums”, J. Number Theory, 59:2 (1996), 398–413 | DOI | MR | Zbl

[19] B. Kerr, “On the congruence $x_1x_2\equiv x_3x_4\,(\operatorname{mod}1)$”, J. Number Theory, 180 (2017), 154–168 | DOI | MR | Zbl

[20] S. M. Voronin, A. A. Karatsuba, Dzeta-funktsiya Rimana, Fizmatlit, M., 1994 | MR

[21] A. Weil, Basic Number Theory, Springer, New York, 1974 | MR | Zbl

[22] P. Erdös, P. Turán, “On a problem in the theory of uniform distribution. I”, Proc. Nederl. Akad. Wetensch., 51 (1948), 1146–1154 | MR | Zbl