Evaluation of the convolution sums ${\sum }_{l+15m=n}\sigma (l)\sigma (m)$ and ${\sum }_{3l+5m=n}\sigma (l)\sigma (m)$ and some applications

Abstract

We evaluate the convolution sums ${\sum }_{l,m\in \mathbb{N},l+15m=n}\sigma (l)\sigma (m)$ and ${\sum }_{l,m\in \mathbb{N},3l+5m=n}\sigma (l)\sigma (m)$ for all $n\in \mathbb{N}$ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer $n$ by the form $$x_1^2+x_1{x}_2+x_2^2+x_3^2+x_3{x}_4+x_4^2+5(x_5^2+x_5{x}_6+x_6^2+x_7^2+x_7{x}_8+x_8^2).$$ We also determine the number of representations of positive integers by the quadratic form $$x_1^2+x_2^2+x_3^2+x_4^2+6(x_5^2+x_6^2+x_7^2+x_8^2),$$ by using the convolution sums obtained earlier by Alaca, Alaca and Williams \cite{{aw3}, {aw4}}.

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