Elementary Evaluation of Convolution Sums involving primitive Dirichlet Characters for a Class of positive Integers

TL;DR

This paper extends the evaluation of convolution sums involving primitive Dirichlet characters for all positive integers, using modular forms, and applies these results to count representations of integers by specific octonary quadratic forms.

Contribution

It generalizes previous results to all positive integers and introduces new methods for evaluating convolution sums using modular forms and Dirichlet characters.

Findings

Evaluated convolution sums for new classes of integers.

Derived formulas for representations by specific octonary quadratic forms.

Provided explicit examples illustrating the method.

Abstract

We extend the results obtained by E. Ntienjem to all positive integers. Let $\EuFrak{N}$ be the subset of $\mathbb{N}$ consisting of $\,2^{\nu}\mho$, where $\nu$ is in $\{0,1,2,3\}$ and $\mho$ is a squarefree finite product of distinct odd primes. We discuss the evaluation of the convolution sum, $\underset{\substack{ {(l,m)\in\mathbb{N}^{2}} {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)$, when $\alpha\beta$ is in $\mathbb{N}\setminus\EuFrak{N}$. The evaluation of convolution sums belonging to this class is achieved by applying modular forms and primitive Dirichlet characters. In addition, we revisit the evaluation of the convolution sums for $\alpha\beta=9$, $16$, $18$, $25$, $36$. If $\alpha\beta\equiv 0 \pmod{4}$, we determine natural numbers $a,b$ and use the evaluated convolution sums together with other known convolution sums to carry out the number of representations of $n$ by the octonary quadratic forms $a\,(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})+ b\,(x_{5}^{2} + x_{6}^{2} + x_{7}^{2} + x_{8}^{2})$. Similarly, if $\alpha\beta\equiv 0 \pmod{3}$, we compute natural numbers $c,d$ and make use of the evaluated convolution sums together with other known convolution sums to determine the number of representations of $n$ by the octonary quadratic forms $c\,(\,x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} + x_{3}^{2} + x_{3}x_{4} + x_{4}^{2}\,) + d\,(\,x_{5}^{2} + x_{5}x_{6} + x_{6}^{2} + x_{7}^{2} + x_{7}x_{8} + x_{8}^{2}\,)$. We illustrate our method with the explicit examples $\alpha\beta = 3^{2}\cdot 5$, $\alpha\beta = 2^{4}\cdot 3$, $\alpha\beta = 2\cdot 5^{2}$ and $\alpha\beta = 2^{6}$, .