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, $\underset{\substack{{(l,m) \in \mathbb{N}_0^2} \\ {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)$, 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.