The main purpose of this paper is to study the mean value properties of a sum analogous to character sums over short intervals by using the mean value theorems for the Dirichlet L-functions, and to give some interesting asymptotic formulae.

Various properties of classical Dedekind sums $S(h,q)$ have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to...

Let $q\ge 3$ be an integer, let $\chi$ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions $$L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac{\chi \left(n\right)}{{(n+a)}^s},$$ where $s=\sigma +\mathrm{i}t$ with $\sigma >1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$-functions especially for $s=1$ and $s=\frac{1}{2}+\mathrm{i}t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.

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 of $$\sum _m\sum _{\chi }\sum _{\overline{\chi }}{\left|C(m,n;k;\chi ;\overline{\chi };q)\right|}^4,$$ and...

In this paper we study the asymptotic behavior of the mean value of Dedekind sums, and give a sharper asymptotic formula.