On a conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series

TL;DR

This paper proves a conjecture relating a double Lambert series to the sum of divisors function by transforming it into a single sum and deriving a new representation of quasi-modular forms.

Contribution

It provides a proof of a recent conjecture connecting double Lambert series with divisor sums, introducing a new representation of quasi-modular forms.

Findings

Confirmed the conjecture relating Lambert series to divisor sums.

Derived a new representation of the quasi-modular form E_2(q).

Transformed a double Lambert series into a single sum for analysis.

Abstract

In this note, we prove a recent conjecture of Amdeberhan, Andrews and Ballantine concerning a double Lambert series. More precisely, they conjectured that \[ \coeff{q^{N2^a}} \sum_{m,k\geq 1} \frac{q^{mk2^a}}{(1+q^{k2^{a-1}})(1-q^{2m-1})} =\sigma_1(N), \] where $\sigma_1(N)$ is the sum of all the positive divisors of $N$. The main idea of the proof is to first transform a double Lambert series on the left-hand side into a single sum. This leads us to derive a new representation of quasi-modular forms $E_2(q)$.