The convolution sum ${\sum }_{al+bm=n}\sigma (l)\sigma (m)$ for $(a,b)=(1,28),(4,7),(1,14),(2,7),(1,7)$

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 $x_1^2+x_2^2+x_3^2+x_4^2+7(x_5^2+x_6^2+x_7^2+x_8^2)$. Finally we compare our evaluations of the sums $W_{1,7}(n)$ and $W_{1,14}(n)$ with the evaluations of Lemire and Williams [10] and Royer [13] to express the modular forms ${\Delta }_{4,7}(z)$, ${\Delta }_{4,14,1}(z)$ and ${\Delta }_{4,14,2}(z)$ (given in [10, 13]) as linear combinations of eta quotients.

13 pages