Evaluation of the convolution sums $W_{1,42}(n),$ $W_{2,21}(n),$ $W_{3,14}(n)$ and $W_{6,7}(n)$

TL;DR

This paper evaluates specific convolution sums using modular forms and applies these results to determine the number of representations of integers by a particular quadratic form.

Contribution

It introduces a modular form approach to explicitly evaluate convolution sums and uses these to analyze representations by a complex quadratic form.

Findings

Explicit formulas for convolution sums involving divisor functions.

Determination of representation counts for a specific quadratic form.

Application of modular forms to quadratic form analysis.

Abstract

In this paper, we use a modular form approach to evaluate the convolution sums $\sum_{l+42m=n}\sigma (l)\sigma (m)$, $\sum_{2l+21m=n}\sigma (l)\sigma (m),$ $\sum_{3l+14m=n}\sigma (l)\sigma (m)$ and $\sum_{6l+7m=n}\sigma (l)\sigma (m) $ for all positive integers $n\mathbf{,}$ and then use their evaluations to determine the number of representation of a positive integer $n$ by the quadratic form $x_{1}^{2} +x_{1}x_{2} +x_{2}^{2} +x_{3}^{2} +x_{3}x_{4} +x_{4}^{2} + 14(x_{5}^{2} +x_{5}x_{6} +x_{6}^{2} +x_{7}^{2} +x_{7}x_{8} +x_{8}^{2})$.