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...

The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan’s sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers $m$, $n$, $k$, $q$, with $k\ge 1$ and $q\ge 3$, and Dirichlet characters $\chi$, $\overline{\chi }$ modulo $q$ we define a mixed exponential sum $$C(m,n;k;\chi ;\overline{\chi };q)=\sum _{a=1}^q{width0ptheight1em}^{\text{'}}\chi \left(a\right)G_k(a,\overline{\chi })e\left(\frac{m{a}^k+n\overline{a^k}}{q}\right),$$ with Dirichlet character $\chi$ and general Gauss sum $G_k(a,\overline{\chi })$ as coefficient, where ${\sum }^{\text{'}}$ denotes the summation over all $a$ such that $(a,q)=1$, $a\overline{a}\equiv 1modq$ and $e\left(y\right)={\mathrm{e}}^{2\pi \mathrm{i}y}$. We mean value...

Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL${}_2$ and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL${}_3$ have been considered, in which analogous GL${}_3$-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL${}_3\left(\mathbb{Z}\right)$. We give formulas...

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.