Let $ q\ge3 $ be a positive integer. For any integers $ m $, $ n $, $ k $, $ h $, the two-term exponential sums $ C(m, n, k, h; q) $ is defined as $ C(m, n, k, h;q) = \sum\limits_{a = 1}^{q}e\left(\frac{ma^k+na^h}{q}\right) $, where $ k > h\ge 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.
Citation: Zhefeng Xu, Xiaoying Liu, Luyao Chen. Hybrid mean value involving some two-term exponential sums and fourth Gauss sums[J]. Electronic Research Archive, 2025, 33(3): 1510-1522. doi: 10.3934/era.2025071
Let $ q\ge3 $ be a positive integer. For any integers $ m $, $ n $, $ k $, $ h $, the two-term exponential sums $ C(m, n, k, h; q) $ is defined as $ C(m, n, k, h;q) = \sum\limits_{a = 1}^{q}e\left(\frac{ma^k+na^h}{q}\right) $, where $ k > h\ge 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.
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Zhefeng Xu, Xiaoying Liu, Luyao Chen. Hybrid mean value involving some two-term exponential sums and fourth Gauss sums[J]. Electronic Research Archive, 2025, 33(3): 1510-1522. doi: 10.3934/era.2025071
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