The convolution sum $\sum_{al+bm=n} \sigma(l) \sigma(m)$ for $(a,b)=(1,28), (4,7), (1,14), (2,7), (1,7)$

TL;DR

This paper evaluates specific convolution sums involving divisor functions using modular forms, re-evaluates known sums, and applies these results to count representations of integers by certain quadratic forms, also relating modular forms to eta quotients.

Contribution

The paper introduces a modular form approach to evaluate convolution sums for specific parameters and re-evaluates known sums, linking them to quadratic form representations and eta quotients.

Findings

Explicit formulas for convolution sums W_{a,b}(n) for given (a,b)

Connections established between convolution sums and representations by quadratic forms

Expressions of modular forms as linear combinations of eta quotients

Abstract

We evaluate the convolution sum $\displaystyle W_{a,b}(n):= \sum_{al+bm=n} \hspace{-3mm} \sigma(l) \sigma(m)$ for $(a,b)=(1,28), (4,7), (2,7)$ for all positive integers $n$. We use a modular form approach. We also re-evaluate the known sums $W_{1,14}(n)$ and $W_{1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of $n$ by the octonary quadratic form $x_1^2 + x_2^2 +x_3^2 + x_4^2 + 7(x_5^2 + x_6^2 + x_7^2 + x_8^2)$. Finally we compare our evaluations of the sums $W_{1,7}(n)$ and $W_{1,14}(n)$ with the evaluations of Lemire and Williams [10] and Royer [13] to express the modular forms $\Delta_{4,7}(z)$, $\Delta_{4,14, 1}(z)$ and $\Delta_{4,14, 2}(z)$ (given in [10, 13]) as linear combinations of eta quotients.