Kyungpook Mathematical Journal

  • 제59권3호
  • /
  • Pages.377-389
  • /
  • 2019
  • /
  • 1225-6951(pISSN)
  • /
  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
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The Convolution Sum $\sum_{al+bm=n}{\sigma}(l){\sigma}(m)$ for (a, b) = (1, 28),(4, 7),(1, 14),(2, 7),(1, 7)

  • Alaca, Ayse (School of Mathematics and Statistics, Carleton University) ;
  • Alaca, Saban (School of Mathematics and Statistics, Carleton University) ;
  • Ntienjem, Ebenezer (School of Mathematics and Statistics, Carleton University)
  • 투고 : 2017.07.18
  • 심사 : 2018.07.18
  • 발행 : 2019.09.23
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초록

We evaluate the convolution sum $W_{a,b}(n):=\sum_{al+bm=n}{\sigma}(l){\sigma}(m)$ for (a, b) = (1, 28),(4, 7),(2, 7) for all positive integers n. We use a modular form approach. We also re-evaluate the known sums $W_{1,14}(n)$ and $W_{1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of n by the octonary quadratic form $x^2_1+x^2_2+x^2_3+x^2_4+7(x^2_5+x^2_6+x^2_7+x^2_8)$. Finally we express the modular forms ${\Delta}_{4,7}(z)$, ${\Delta}_{4,14,1}(z)$ and ${\Delta}_{4,14,2}(z)$ (given in [10, 14]) as linear combinations of eta quotients.

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과제정보

연구 과제 주관 기관 : Natural Sciences and Engineering Research Council of Canada

참고문헌

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