Geodesic
Kloosterman sums with multiplicative coefficients
Izvestiya. Mathematics , Tome 82 (2018) no. 4, pp. 647-661.

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We obtain several new bounds for sums of the form $$ S_{q}(x;f)=\mathop{{\sum}'}_{n\le x}f(n)e_{q}(an^*+bn), $$ in which $q$ is a sufficiently large integer, $\sqrt{q}\,(\log{q})\ll x\le q$, $a$ and $b$ are integers with $(a,q)=1$, $e_{q}(v) = e^{2\pi iv/q}$, $f(n)$ is a multiplicative function satisfying certain conditions, $nn^*\equiv 1 \pmod{q}$, and the prime in the sum means that $(n,q)=1$. The results in this paper refine similar bounds obtained earlier by Gong and Jia.
Keywords: Kloosterman sums, multiplicative functions.
Mots-clés : inverse residues 
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M. A. Korolev. Kloosterman sums with multiplicative coefficients. Izvestiya. Mathematics , Tome 82 (2018) no. 4, pp. 647-661. http://geodesic.mathdoc.fr/item/IM2_2018_82_4_a0/

[1] M. A. Korolev, “Incomplete Kloosterman sums and their applications”, Izv. Math., 64:6 (2000), 1129–1152 | DOI | DOI | MR | Zbl

[2] M. A. Korolev, “Short Kloosterman sums with weights”, Math. Notes, 88:3 (2010), 374–385 | DOI | DOI | MR | Zbl

[3] M. A. Korolev, “Karatsuba's method for estimating Kloosterman sums”, Sb. Math., 207:8 (2016), 1142–1158 | DOI | DOI | MR | Zbl

[4] M. A. Korolev, “On short Kloosterman sums modulo a prime”, Math. Notes, 100:6 (2016), 820–827 | DOI | DOI | MR | Zbl

[5] M. A. Korolev, “Short Kloosterman sums to powerful modulus”, Dokl. Math., 94:2 (2016), 561–562 | DOI | DOI | MR | Zbl

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

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

[8] M. A. Korolev, “Generalized Kloosterman sum with primes”, Proc. Steklov Inst. Math., 296 (2017), 154–171 | DOI | DOI | MR | Zbl

[9] M. A. Korolev, “On a Diophantine inequality with reciprocals”, Proc. Steklov Inst. Math., 299 (2017), 132–142 | DOI | DOI

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

[11] S. V. Konyagin, M. A. Korolev, “On a symmetric Diophantine equation with reciprocals”, Proc. Steklov Inst. Math., 294 (2016), 67–77 | DOI | DOI | MR | Zbl

[12] S. V. Konyagin, M. A. Korolev, “Irreducible solutions of an equation involving reciprocals”, Sb. Math., 208:12 (2017), 1818–1834 | DOI | DOI | MR | Zbl

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

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

[15] S. A. Stepanov, I. E. Shparlinskii, “Ob otsenke trigonometricheskikh summ s ratsionalnymi i algebraicheskimi funktsiyami”, Avtomorfnye funktsii i teoriya chisel, v. 1, DVO AN SSSR, Vladivostok, 1989, 5–18 | MR | Zbl

[16] A. A. Karatsuba, “The distribution of inverses in a residue ring modulo a given modulus”, Russian Acad. Sci. Dokl. Math., 48:3 (1994), 452–454 | MR | Zbl

[17] A. A. Karatsuba, “Fractional parts of functions of a special form”, Izv. Math., 59:4 (1995), 721–740 | DOI | MR | Zbl

[18] A. A. Karatsuba, “Analogues of Kloosterman sums”, Izv. Math., 59:5 (1995), 971–981 | DOI | MR | Zbl

[19] A. A. Karatsuba, “Sums of fractional parts of functions of a special form”, Dokl. Math., 54:1 (1996), 541 | MR | Zbl

[20] A. A. Karatsuba, “Analogi nepolnykh summ Kloostermana i ikh prilozheniya”, Tatra Mt. Math. Publ., 11 (1997), 89–120 | MR | Zbl

[21] A. A. Karatsuba, “Kloosterman double sums”, Math. Notes, 66:5 (1999), 565–569 | DOI | DOI | MR | Zbl

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

[23] Ke Gong, ChaoHua Jia, “Kloosterman sums with multiplicative coefficients”, Sci. China Math., 59:4 (2016), 653–660 ; arXiv: 1401.4556v4 | DOI | MR | Zbl

[24] Peiming Deng, “On Kloosterman sums with oscillating coefficients”, Canad. Math. Bull., 42:3 (1999), 285–290 | DOI | MR | Zbl

[25] D. Hajela, A. Pollington, B. Smith, “On Kloosterman sums with oscillating coefficients”, Canad. Math. Bull., 31:1 (1988), 32–36 | DOI | MR | Zbl

[26] Chinese J. Contemp. Math., 19:2 (1998), 185–191 | MR | Zbl

[27] 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

[28] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Stud. Adv. Math., 46, Cambridge Univ. Press, Cambridge, 1995, xvi+448 pp. | MR | Zbl

[29] J. B. Rosser, L. Schoenfeld, “Approximate formulas for some functions of prime numbers”, Illinois J. Math., 6 (1962), 64–94 | MR | Zbl

[30] K. Prachar, Primzahlverteilung, Grundlehren Math. Wiss., 91, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1957, x+415 pp. | MR | MR | Zbl | Zbl

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

[32] A. A. Karatsuba, S. M. Voronin, The Riemann zeta-function, De Gruyter Exp. Math., 5, Walter de Gruyter Co., Berlin, 1992, xii+396 pp. | DOI | MR | MR | Zbl | Zbl