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 
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/
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     author = {N. V. Proskurin},
     title = {On cubic exponential sums and {Gauss} sums},
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     pages = {159--163},
     year = {2017},
     volume = {458},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_458_a8/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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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