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

TL;DR

This paper evaluates convolution sums involving the sum of divisors function for specific parameters, generalizes the method using Eisenstein forms, and applies results to count representations of integers by certain quadratic forms.

Contribution

It introduces a generalized approach to evaluate convolution sums with Eisenstein forms and derives formulas for representing integers by specific octonary quadratic forms.

Findings

Derived explicit formulas for convolution sums with αβ=14,22,26.

Established connections between convolution sums and quadratic form representations.

Extended existing methods to a broader class of convolution sums.

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.