Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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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)
  • Received : 2012.06.19
  • Accepted : 2012.08.20
  • Published : 2013.01.30
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Abstract

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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References

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