Elementary Evaluation of Convolution Sums involving the Sum of Divisors Function for a Class of positive Integers

TL;DR

This paper introduces an elementary method using modular forms to evaluate convolution sums involving the sum of divisors function for certain positive integers, and applies these results to count representations by octonary quadratic forms.

Contribution

It provides a new elementary approach to evaluate convolution sums for specific integer classes and extends the application to counting representations by octonary quadratic forms.

Findings

Explicit evaluation of convolution sums for specific products of  and 

Derived formulas for the number of representations of integers by octonary quadratic forms

Generalized method applicable to all natural numbers for convolution sum extraction

Abstract

We discuss an elementary method for the evaluation of the convolution sums $\underset{\substack{ {(l,m)\in\mathbb{N}_{0}^{2}} \\ {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)$ for those $\alpha,\beta\in\mathbb{N}$ for which $\gcd{(\alpha,\beta)}=1$ and $\alpha\beta=2^{\nu}\mho$, where $\nu\in\{0,1,2,3\}$ and $\mho$ is a finite product of distinct odd primes. Modular forms are used to achieve this result. We also generalize the extraction of the convolution sum to all natural numbers. Formulae for the number of representations of a positive integer $n$ by octonary quadratic forms using convolution sums belonging to this class are then determined when $\alpha\beta\equiv 0\pmod{4}$ or $\alpha\beta\equiv 0\pmod{3}$. To achieve this application, we first discuss a method to compute all pairs $(a,b),(c,d)\in\mathbb{N}^{2}$ necessary for the determination of such formulae for the number of representations of a positive integer $n$ by octonary quadratic forms when $\alpha\beta$ has the above form and $\alpha\beta\equiv 0\pmod{4}$ or $\alpha\beta\equiv 0\pmod{3}$. We illustrate our approach by explicitly evaluating the convolution sum for $\alpha\beta=33=3\cdot 11,\> \alpha\beta=40=2^{3}\cdot 5$ and $\alpha\beta=56=2^{3}\cdot 7$, and by revisiting the evaluation of the convolution sums for $\alpha\beta=10$, $11$, $12$, $15$, $24$. We then apply these convolution sums to determine formulae for the number of representations of a positive integer $n$ by octonary quadratic forms. In addition, we determine formulae for the number of representations of a positive integer $n$ when $(a,b)=(1,1)$, $(1,3)$, $(2,3)$, $(1,9)$.