Let $q$ be a positive integer, $\chi$ denote any Dirichlet character $modq$. For any integer $m$ with $(m,q)=1$, we define a sum $C(\chi ,k,m;q)$ analogous to high-dimensional Kloosterman sums as follows: $$C(\chi ,k,m;q)=\sum _{a_1=1}^q{}^{\text{'}}\sum _{a_2=1}^q{}^{\text{'}}\cdots \sum _{a_k=1}^q{}^{\text{'}}\chi (a_1+a_2+\cdots +a_k+m\overline{a_1{a}_2\cdots a_k}),$$ where $a·\overline{a}\equiv 1modq$. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value $\left|C\right(\chi ,k,m;q\left)\right|$, and give two interesting identities for it.

For the general modulo $q\ge 3$ and a general multiplicative character $\chi$ modulo $q$, the upper bound estimate of $\left|S\right(m,n,1,\chi ,q\left)\right|$ is a very complex and difficult problem. In most cases, the Weil type bound for $\left|S\right(m,n,1,\chi ,q\left)\right|$ is valid, but there are some counterexamples. Although the value distribution of $\left|S\right(m,n,1,\chi ,q\left)\right|$ is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for $k$-th Kloosterman sums and analytic method to study the asymptotic...

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 $C(m,n,k;q)={\sum }_{a=1}^{q}e((m{a}^k+na)/q)$, where $e\left(y\right)={\mathrm{e}}^{2\pi \mathrm{i}y}$. 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.