Evaluation of the Convolution Sum involving the Sum of Divisors Function for 14, 22 and 26

Abstract

For all natural numbers $n$, we discuss the evaluation of the convolution sum, $\sum _{\begin{array}{c}(l,m)\in {\mathbb{N}}_0^2\\ \alpha l+\beta m=n\end{array}}\sigma (l)\sigma (m)$, where $\alpha \beta =14,22,26$. We generalize the extraction of the convolution sum using Eisenstein forms of weight $4$ for all pairs of positive integers $(\alpha ,\beta )$. We also determine formulae for the number of representations of a positive integer by the octonary quadratic forms $a(x_1^2+x_2^2+x_3^2+x_4^2)+b(x_5^2+x_6^2+x_7^2+x_8^2)$, where $(a,b)=(1,1),(1,3),(2,3),(1,9)$. These numbers of representations of a positive integer are applications of the evaluation of certain convolution sums by J. G. Huard et al., A. Alaca et al. and D. Ye.

13 pages