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A hybrid mean value involving two-term exponential sums and polynomial character sums
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 53-62
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Let $q \ge 3$ be a positive integer. For any integers $m$ and $n$, the two-term exponential sum $C(m,n,k;q)$ is defined by $C(m,n,k;q) = \sum _{a=1}^q e ({(ma^k +na)}/{q})$, where $e(y)={\rm e}^{2\pi {\rm i} y}$. In this paper, we use the properties of Gauss sums and the estimate for Dirichlet character of polynomials to study the mean value problem involving two-term exponential sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it.
Let $q \ge 3$ be a positive integer. For any integers $m$ and $n$, the two-term exponential sum $C(m,n,k;q)$ is defined by $C(m,n,k;q) = \sum _{a=1}^q e ({(ma^k +na)}/{q})$, where $e(y)={\rm e}^{2\pi {\rm i} y}$. In this paper, we use the properties of Gauss sums and the estimate for Dirichlet character of polynomials to study the mean value problem involving two-term exponential sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it.
DOI : 10.1007/s10587-014-0082-0
Classification : 11F20, 11L40
Keywords: Dirichlet character of polynomials; two-term exponential sums; hybrid mean value; asymptotic formula 
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Di, Han. A hybrid mean value involving two-term exponential sums and polynomial character sums. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 53-62. doi: 10.1007/s10587-014-0082-0

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