Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
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APPLICATION OF CONVOLUTION SUM ∑k=1N-1σ1(k)σ1(2nN-2nk)

  • Kim, Daeyeoul (National Institute for Mathematical Sciences) ;
  • Kim, Aeran (Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University)
  • 투고 : 2012.06.19
  • 심사 : 2012.08.20
  • 발행 : 2013.01.30
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초록

Let $$S^{\pm}_{(n,k)}\;:=\{(a,b,x,y){\in}\mathbb{N}^4:ax+by=n,x{\equiv}{\pm}y\;(mod\;k)\}$$. From the formula $\sum_{(a,b,x,y){\in}S^{\pm}_{(n,k)}}\;ab=4\sum_{^{m{\in}\mathbb{N}}_{m<n/k}}\;{\sigma}_1(m){\sigma}_1(n-km)+\frac{1}{6}{\sigma}_3(n)-\frac{1}{6}{\sigma}_1(n)-{\sigma}_3(\frac{n}{k})+n{\sigma}_1(\frac{n}{k})$, we find the Diophantine solutions for modulo $2^{m^{\prime}}$ and $3^{m^{\prime}}$, where $m^{\prime}{\in}\mathbb{N}$.

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참고문헌

  1. J. G. Huard, Z. M. Ou, B. K. Spearman, and K. S. Williams, Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions, Number theory for the millennium, II, (2002), 229-274.
  2. D. Kim, A. Kim, J. Kim, and H. Cho, A new approach to tree model from convolution sums of divisor functions modulo 3, Submitted.
  3. D. Kim, A. Kim, and A. Sankaranarayanan, Some properties of products of convolution sums, Submitted.
  4. K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society, Student Texts 76, Cambridge, (2011).