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 
@article{IM2_2018_82_4_a0,
     author = {M. A. Korolev},
     title = {Kloosterman sums with multiplicative coefficients},
     journal = {Izvestiya. Mathematics },
     pages = {647--661},
     publisher = {mathdoc},
     volume = {82},
     number = {4},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2018_82_4_a0/}
}
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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/