Mean values of the homogeneous Dedekind sums
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Let $a$, $b$, $q$ be integers with $q>0$. The homogeneous Dedekind sum is defined by $$ S(a,b,q)=\sum_{r=1}^q\left(\left(\frac{ar}{q}\right)\right)\left(\left(\frac{br}{q}\right)\right), $$ where $$ ((x))=\left\{\begin{array}{ll} x-[x]-\frac{1}{2}, & \hbox{if $x$ is not an integer},\\ 0, & \hbox{if $x$ is an integer}. \end{array} \right. $$ In this paper we study the mean value of $S(a,b,q)$ by using mean value theorems of Dirichlet $L$-functions, and give some asymptotic formula.
Abstract
Let $a$, $b$, $q$ be integers with $q>0$. The homogeneous Dedekind sum is defined by $$ S(a,b,q)=\sum_{r=1}^q\left(\left(\frac{ar}{q}\right)\right)\left(\left(\frac{br}{q}\right)\right), $$ where $$ ((x))=\left\{\begin{array}{ll} x-[x]-\frac{1}{2}, & \hbox{if $x$ is not an integer},\\ 0, & \hbox{if $x$ is an integer}. \end{array} \right. $$ In this paper we study the mean value of $S(a,b,q)$ by using mean value theorems of Dirichlet $L$-functions, and give some asymptotic formula.
참고문헌 (9)
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