Evaluation of the Convolution Sums $\underset{\substack{ {(l,m)\in\mathbb{N}_{0}^{2}} {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)$, where $\alpha\beta=44,52$

TL;DR

This paper evaluates specific convolution sums involving divisor functions for certain parameters and applies these results to derive formulas for counting representations of natural numbers by particular octonary quadratic forms.

Contribution

It provides explicit evaluations of convolution sums for \\alpha\\beta=44,52 and uses them to find formulas for representations by specific octonary quadratic forms.

Findings

Explicit formulas for convolution sums with \\alpha\\beta=44,52.

Derived formulas for counting representations by quadratic forms.

Connections established between convolution sums and quadratic form representations.

Abstract

The convolution sum, $\underset{\substack{ {(l,m)\in\mathbb{N}_{0}^{2}} {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)$, where $\alpha\beta=44,52$, is evaluated for all natural numbers $n$. We then use these convolution sums to determine formulae for the number of representations of a natural number 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 $(a,b)= (1,11),(1,13)$.