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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 2017 Volume 458, Pages 159–163 (Mi znsl6457)

On cubic exponential sums and Gauss sums

N. V. Proskurin

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

Abstract: 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 $n\in\mathbb F_q$.

Key words and phrases: Gauss sum, finite field, cubic exponential sum.

UDC: 511.321

Received: 13.09.2017


 English version:
Journal of Mathematical Sciences (New York), 2018, 234:5, 697–700


© Steklov Math. Inst. of RAS, 2025