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

TL;DR

This paper evaluates specific convolution sums involving divisor functions using quasimodular forms and applies these results to count representations of integers by certain quadratic forms.

Contribution

It provides explicit evaluations of convolution sums for all natural numbers and applies them to determine the number of representations by complex quadratic forms.

Findings

Explicit formulas for convolution sums involving divisor functions.

Number of representations of integers by specific quadratic forms.

Application of quasimodular forms in convolution sum evaluation.

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 $$ x_1^2 + x_1x_2 + x_2^2 + x_3^2 + x_3x_4 + x_4^2 + 5 (x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2). $$ 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}}.