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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.
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참고문헌 (9)
참고문헌 신청
H. Liu / 2015 / On the mean values of Dirichlet L-functions / Journal of Number Theory 147 : 172 ~ 183
H. Rademacher / 1972 / Carus Mathematical Monographs / Mathemat-ical Association of America
Huaning Liu / 2007 / On the mean values of the homogeneous Dedekind sums and Cochrane sums in short intervals / 대한수학회지 44 (6) : 1243 ~ 1254
J. Szmidt / 1995 / Congruences among generalized Bernoulli numbers / Acta Arithmetica 71 : 273 ~ 278
R. R. Hall / 1993 / Dedekind sums and continued fractions / Acta Arithmetica 63 : 79 ~ 90
H. Rademacher / 1972 / Carus Mathematical Monographs / Mathemat-ical Association of America
Huaning Liu / 2007 / On the mean values of the homogeneous Dedekind sums and Cochrane sums in short intervals / 대한수학회지 44 (6) : 1243 ~ 1254
J. Szmidt / 1995 / Congruences among generalized Bernoulli numbers / Acta Arithmetica 71 : 273 ~ 278
R. R. Hall / 1993 / Dedekind sums and continued fractions / Acta Arithmetica 63 : 79 ~ 90
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