Mean values of the homogeneous Dedekind sums
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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.
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References
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