Polylogarithms and the asymptotic formula for the moments of Lebesgue’s singular function

Recall the Lebesgue's singular function. We define a Lebesgue's singular function $L(t)$ as the unique continuous solution of the functional equation
$$ L(t) = qL(2t) +pL(2t-1), $$
where $p,q>0$, $q=1-p$, $p\ne q$. The moments of Lebesque' singular function are defined as
$$ M_n = \int_0^1t^n dL(t), \quad n = 0, 1, \dots $$
The main result of this paper is
$$ M_n = n^{\log_2 p} e^{-\tau(n)}\left(1 + \mathcal{O}(n^{-0.99})\right), $$
where
\begin{gather*} \tau(x) = \frac12\ln p + \Gamma'(1)\log_2 p +\frac1{\ln 2}\frac{\partial}{\partial z}\left.\mathrm{Li}_{z}\left(-\frac{q}{p}\right)\right|_{z=1} +\frac1{\ln 2}\sum_{k\ne0} \Gamma(z_k)\mathrm{Li}_{z_k+1}\left(-\frac{q}{p}\right) x^{-z_k},\\ z_k = \frac{2\pi ik}{\ln 2}, \ k\ne 0. \end{gather*}
The proof is based on analytic techniques such as the poissonization and the Mellin transform.