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http://dx.doi.org/10.5666/KMJ.2019.59.3.377

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)
Publication Information
Kyungpook Mathematical Journal / v.59, no.3, 2019 , pp. 377-389 More about this Journal
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;
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