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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)
  • Received : 2017.07.18
  • Accepted : 2018.07.18
  • Published : 2019.09.23

Abstract

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.

Keywords

convolution sums;sum of divisors function;Eisenstein series;modular forms;cusp forms;Dedekind eta function;eta quotients;octonary quadratic forms;representations

Acknowledgement

Supported by : Natural Sciences and Engineering Research Council of Canada

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