This is a cross-post of the same math.SE question to MO, thinking that is better suited here.
My question is about Lemma 2.1 from the ArXiv:1812.07690 by D. Zagier and S. Garoufalidis which concerns q-pochhammer symbol $(z; q)_{\infty} = \prod_{k=0}^{\infty} \left(1 - z \cdot q^k \right)$, an entire function of $z$ for all complex $q$ such that $\left\vert q \right\vert < 1$.
Consider $q$ to be near the root of unity: $q = \zeta \exp\left(-\epsilon/m\right)$, where $\zeta$ is the $m$-th primitive root of unity, $\epsilon$ is small, and $z = q \ w \ \exp\left( -\nu \epsilon/m\right)$ for a complex $\nu$, such that $\nu \ \epsilon = \mathcal{o}(1)$ the lemma 2.1 states
$$\begin{align} -\log\left(q\ w \ e^{-\nu \ \epsilon/m}; q\right)_{\infty} &= \left.- \sum_{n=1}^\infty \log \left(1- q^n w e^{-\nu \epsilon/m}\right) \right|_{\mathrm{using}\ \ -\log(1-u) = \sum_{k=1}^\infty \frac{u^k}{k} }\\ &= \left. \sum_{k=1}^\infty \frac{1}{k} \sum_{n=1}^\infty q^{n k} w^k e^{-\nu k \epsilon/m} \right|_{\sum_{n=1}^\infty q^{n k} = \sum_{t=1}^{m} \sum_{s=0}^\infty q^{t k} q^{m k s} = \sum_{t=1}^m \frac{q^{t k}}{1-q^{m k}} } \\ &= \left.\sum_{k=1}^\infty \frac{1}{k} \sum_{t=1}^{m} \left(\zeta^t w\right)^k \frac{e^{-(\nu +t) \epsilon\ k/m} }{1 - e^{- \epsilon \ k}}\right\vert_{\frac{e^{-(\nu +t) \epsilon\ k/m} }{1 - e^{- \epsilon \ k}}=\sum_{r=0}^\infty \frac{ \left(k \epsilon\right)^{r-1}}{r!} B_{r}\left(1-\frac{\nu+t}{m}\right) } \\ &= \left.\sum_{r=0}^\infty \frac{\epsilon^{r-1}}{r!} \sum_{t=1}^m B_{r}\left(1-\frac{\nu+t}{m}\right) \sum_{k=1}^\infty \frac{\left(\zeta^t w\right)^k}{k} k^{r-1} \right\vert_{ \mathrm{Li}_{s}(w) = \sum_{k=1}^\infty \frac{z^k}{k^s}} \\ &= \sum_{r=0}^\infty \frac{\epsilon^{r-1}}{r!} \sum_{t=1}^m B_{r}\left(1-\frac{\nu+t}{m}\right) \mathrm{Li}_{2-r}\left(\zeta^t w\right) \end{align} $$
This sequence clearly demonstrates that the expansion is $\mathbb{C}[\nu][[\epsilon]]$, i.e. polynomial in $\nu$ and containing some negative powers of $\epsilon$.
In addition to this, the lemma 2.1 states $\psi_{w, \zeta}\left(\nu,\epsilon\right)$ defined as $$ \psi_{w, \zeta}\left(\nu,\epsilon\right) \left.=\right._{\mathrm{def}} \sum_{r=2}^\infty \frac{\epsilon^{r-1}}{r!} \sum_{t=1}^m B_{r}\left(1-\frac{\nu+t}{m}\right) \mathrm{Li}_{2-r}\left(\zeta^t w\right) $$ has an explicit asymptotic expansion in $\mathbb{C}[\nu \epsilon^{2/3}][[ \epsilon^{1/3} ]] \subset \mathbb{C}[\nu][[\epsilon]]$ as $\epsilon \searrow 0$. I fail to see why this claim might be true, and how one would derive this explicit asymptotic expansion.
I am hoping to get help with that here.
