The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

中文翻译：

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对于 k = a · b = 21、33 和 35 的卷积和求值

文章重点介绍卷积和的评估 $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ 涉及除数之和的函数 $$\sigma (n)$$ 为了ķ=21、33和35。在本文中，我们的目标是获得21级的某些爱森斯坦级数，并用它们来评估21级的卷积和。我们还利用现有的33级和35级的爱森斯坦级数恒等式在评估级别 33 和 35 的卷积和时。大多数卷积和是使用模形式理论评估的，而我们设计了一种不受模形式理论的技术。作为一个应用程序，我们确定一个正整数的表示数量的公式n由八次方形式 $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ ， 为了 （一，乙)=(1, 7), (1, 11), (2, 3) 和 (2, 5)。