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.

Let $p$ be an odd prime and $c$ a fixed integer with $(c,p)=1$. For each integer $a$ with $1\le a\le p-1$, it is clear that there exists one and only one $b$ with $0\le b\le p-1$ such that $ab\equiv c$ (mod $p$). Let $N(c,p)$ denote the number of all solutions of the congruence equation $ab\equiv c$ (mod $p$) for $1\le a$, $b\le p-1$ in which $a$ and $\overline{b}$ are of opposite parity, where $\overline{b}$ is defined by the congruence equation $b\overline{b}\equiv 1(modp)$. The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet $L$-functions to study the hybrid mean value...

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...

For $1\le c\le p-1$, let $E_1,E_2,\cdots ,E_m$ be fixed numbers of the set $\{0,1\}$, and let $a_1,a_2,\cdots ,a_m$ $(1\le a_i\le p$, $i=1,2,\cdots ,m)$ be of opposite parity with $E_1,E_2,\cdots ,E_m$ respectively such that $a_1{a}_2\cdots a_m\equiv c(modp)$. Let $$N(c,m,p)=\frac{1}{2^{m-1}}\underset{a_1{a}_2\cdots a_m\equiv c(modp)}{\sum _{a_1=1}^{p-1}\sum _{a_2=1}^{p-1}\cdots \sum _{a_m=1}^{p-1}}(1-{(-1)}^{a_1+E_1})(1-{(-1)}^{a_2+E_2})\cdots (1-{(-1)}^{a_m+E_m}).$$ We are interested in the mean value of the sums $$\sum _{c=1}^{p-1}{E}^2(c,m,p),$$ where $E(c,m,p)=N(c,m,p)-\left({(p-1)}^{m-1}\right)/\left(2^{m-1}\right)$ for the odd prime $p$ and any integers $m\ge 2$. When $m=2$, $c=1$, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.

Let k ≥ 1 denote any positive rational integer. We give formulae for the sums $S_{odd}(k,f)={\sum }_{\chi (-1)=-1}\left|L(k,\chi )\right|²$ (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums $S_{even}(k,f)={\sum }_{\chi (-1)=+1}\left|L(k,\chi )\right|²$ (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.

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.