Evaluation of the Convolution Sums
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For all positive integers $n$ we evaluate the convolution sums $\displaystyle\overset{}{\underset{ \begin{array}{c} {(l,m) \in \mathbb{N}^2} \\ {\alpha l+\beta m=n} \end{array} } {\sum}}\sigma(l)\sigma(m)$, where $(\alpha,\beta) = (1,14), (2,7), (1,26), (2,13), (1,28)$, $(4,7), (1,30), (2,15), (3,10), (5,6)$. Using some of the evaluations of these convolution sums we determine formulae for the number of representations of $n$ by the octonary quadratic forms \begin{equation*} x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}+ 7 (x_{5}^{2} + x_{6}^{2} + x_{7}^{2} + x_{8}^{2}) \end{equation*} and \begin{equation*} a(\,x_{1}^{2} + x_{2}^{2} + x_{1}x_{2} + x_{3}^{2} + x_{3}x_{4} + x_{4}^{2}\,) + b (\,x_{5}^{2} + x_{5}x_{6} + x_{6}^{2} + x_{7}^{2} + x_{7}x_{8} + x_{8}^{2}\,), \end{equation*} where $(a,b)$ stands for $(1, 10)$ or $(2,5)$.
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Copyright © 2015 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, research, scholarship, and teaching. Theses may only be shared by linking to Carleton University Institutional Repository and no part may be used without proper attribution to the author. No part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner.
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