| 摘要: |
| 设$f, k$为正整数，$h$为任意整数，满足$(fk,h)=1$与$(f,k)=1$.设$\overline{B}_n(x)$为$n$次Bernoulli 周期函数,即 $$ \overline{B}_n(x)=\left\{\begin{array}{ll} B_n\left(x-[x]\right), & \hbox{当\ $x$ 不是整数}, \\ 0, & \hbox{当\ $x$ 是整数}. \end{array}\right. $$ 此外设 $\chi$ 为模\ $f$的原特征, $\overline{B}_{n,\chi}(x)$ 为\ $n$ 次广义\ Bernoulli特征函数，满足 $$ \overline{B}_{n,\chi}(x)=\left\{\begin{array}{ll} B_{n,\chi}\left(x-[x]\right), & \hbox{当\ $x$不是整数}, \\ 0, & \hbox{当\ $x$是整数}. \end{array}\right. $$ 定义新型\ Dedekind和如下: $$ s(h, k; \chi)=\sum_{a=0}^{fk-1}\overline{B}_{1,\chi}\left(\frac{ha}{k}\right) \overline{B}_{1}\left(\frac{a}{fk}\right). $$ 本文建立了\ $s(h, k; \chi)$ 与\ Dirichlet $L$-函数的均值之间的关系，并由此给出了\ $s(h, k; \chi)$ 的混合均值的渐近公式. |
| 关键词: 新型Dedekind，Dirichlet L-函数，混合均值. |
| DOI： |
| 分类号:O156.4. |
| 基金项目:国家自然科学基金项目（面上项目，重点项目，重大项目） |
|
| ON THE HYBRID MEAN OF NEW DEDEKIND SUMS |
|
WANG Xiao-ying
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|
Northwest University
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| Abstract: |
| Let $f, k$ be positive integers, $h$ be arbitrary integer, satisfying $(fk,h)=1$ and $(f,k)=1$. Let $\overline{B}_n(x)$ be the $n$-th Bernoulli periodic function such that $$\overline{B}_n(x)=\left\{\begin{array}{ll} B_n\left(x-[x]\right), & \hbox{if\ $x$ is not an integer}, \\ 0, & \hbox{if\ $x$ is an integer}.\end{array}\right.$$ Let $\chi$ be a Dirichlet primitive character modulo $f$, and let $\overline{B}_{n,\chi}(x)$ denote the $n$-th generalized Bernoulli character function satisfying$$ \overline{B}_{n,\chi}(x)=\left\{\begin{array}{ll}B_{n,\chi}\left(x-[x]\right), & \hbox{if\ $x$ is not an integer}, \\ 0, & \hbox{if\ $x$ is an integer}.\end{array}\right.$$ Define new Dedekind sums as following:$$s(h, k;\chi)=\sum_{a=0}^{fk-1}\overline{B}_{1,\chi}\left(\frac{ha}{k}\right) \overline{B}_{1}\left(\frac{a}{fk}\right).$$ This paper established some connection between $s(h, k; \chi)$ and Dirichlet $L$-function,and gave asymptotic formula for the hybrid mean of $s(h, k; \chi)$. |
| Key words: New Dedekind sum Dirichlet $L$-function hybrid mean. |
