Geodesic
On the $k$-polygonal numbers and the mean value of Dedekind sums
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 409-415
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For any positive integer $k\geq 3$, it is easy to prove that the \mbox {$k$-polygonal} numbers are $a_n(k)= (2n+n(n-1)(k-2))/2$. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet \mbox {$L$-functions} and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S(a_n(k)\overline {a}_m(k), p)$ for \mbox {$k$-polygonal} numbers with $1\leq m,n\leq p-1$, and give an interesting computational formula for it.
For any positive integer $k\geq 3$, it is easy to prove that the \mbox {$k$-polygonal} numbers are $a_n(k)= (2n+n(n-1)(k-2))/2$. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet \mbox {$L$-functions} and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S(a_n(k)\overline {a}_m(k), p)$ for \mbox {$k$-polygonal} numbers with $1\leq m,n\leq p-1$, and give an interesting computational formula for it.
DOI : 10.1007/s10587-016-0264-z
Classification : 11L05, 11L10
Keywords: Dedekind sums; mean value; computational problem; $k$-polygonal number; analytic method 
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Guo, Jing; Li, Xiaoxue. On the $k$-polygonal numbers and the mean value of Dedekind sums. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 409-415. doi: 10.1007/s10587-016-0264-z

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