Geodesic
On cubic exponential sums and Gauss sums
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 159-163
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Let $e_q$ be a nontrivial additive character of a finite field $\mathbb F_q$ of order $q\equiv1\pmod3$, and let $\psi$ be a cubic multiplicative character of $\mathbb F_q$, $\psi(0)=0$. Consider the cubic Gauss sum and the cubic exponential sum \begin{equation*} G(\psi)=\sum_{z\in\mathbb F_q}e_q(z)\psi(z),\quad C(w)=\sum_{z\in\mathbb F_q}e_q\Bigl(\frac{z^3}w-3z\Bigr),\quad w\in\mathbb F_q\quad w\neq0. \end{equation*} For all nonzero $a,b\in\mathbb F_q$, $ab\neq0$, it is proved that \begin{equation*} \frac1q\sum_nC(an)C(bn)\psi(n)+\frac1q\psi(ab)G(\psi)^2=\bar\psi(ab)\psi(a-b)\overline{G(\psi)}, \end{equation*} where summation runs over all nonzero $n\in\mathbb F_q$. 
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     title = {On cubic exponential sums and {Gauss} sums},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_458_a8/}
}
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N. V. Proskurin. On cubic exponential sums and Gauss sums. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 159-163. http://geodesic.mathdoc.fr/item/ZNSL_2017_458_a8/

[1] “Convolutions of twisted Kloosterman sums”, Journal of Mathematical Sciences, 129:3 (2005), 3868–3873 | DOI | MR | Zbl

[2] W. Duke, H. Iwaniec, “A relation between cubic exponential and Kloosterman sums”, Contemporary Mathematics, 143 (1993), 255–258 | DOI | MR | Zbl

[3] J. Booher, A. Etropolski, A. Hittson, “Evaluations of cubic twisted Kloosterman sheaves”, International J. Number Theory, 6 (2010), 1349–1365 | DOI | MR | Zbl

[4] K. Ireland, M. Rosen, A classical introduction to modern number theory, Grad. Texts in Math., 84, Second ed., Springer-Verlag, 1990 | DOI | MR | Zbl

[5] D. J. Wright, “Cubic character sums of cubic polynomials”, Proc. Amer. Math. Soc., 100:3 (1987), 409–413 | DOI | MR | Zbl