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

We extend the results obtained by E. Ntienjem to all positive integers. Let $\EuFrak{N}$ be the subset of $\mathbb{N}$ consisting of $\,2^ν\mho$, where $ν$ 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, $\underset{\substack{ {(l,m)\in\mathbb{N}^{2}} {α\,l+β\,m=n} } }{\sum}σ(l)σ(m)$, when $αβ$ 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 $αβ=9$, $16$, $18$, $25$, $36$. If $αβ\equiv 0 \pmod{4}$, 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 $αβ\equiv 0 \pmod{3}$, 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 $αβ= 3^{2}\cdot 5$, $αβ= 2^{4}\cdot 3$, $αβ= 2\cdot 5^{2}$ and $αβ= 2^{6}$, .

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