Hybrid mean value involving some two-term exponential sums and fourth Gauss sums

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Abstract

Let q ≥ 3 be a positive integer. For any integers m, n, k, h, the two-term exponential sums $C(m, n, k, h ; q)=\sum_{a=1}^{q} e\left(\frac{m a^{k}+n a^{h}}{q}\right)$, where k > h ≥ 2. The main purpose of this paper is to use analytic methods and the properties of classical Gauss sums to study the mean value involving two-term exponential sums and fourth Gauss sums, and to provide some asymptotic formulas and identities. Previously, only the case of h = 1 had been studied.

Keywords

two-term exponential sums / fourth Gauss sums / mean value / asymptotic formulas / identities

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