Asymptotic formula for the moments of Takagi function

Takagi function is a simple example of a continuous but nowhere differentiable function. It is defined by
$$ T(x) = \sum_{k=0}^{\infty}2^{-n}\rho(2^nx), $$
where
$$ \rho(x) = \min_{k\in \mathbb{Z}}|x-k|. $$
The moments of Takagi function are defined as
$$ M_n = \int_0^1\,x^n T(x)\,dx. $$
The main result of this paper is the following:
$$ M_n = \frac{\ln n - \Gamma'(1)-\ln\pi}{n^2\ln 2}+\frac{1}{2n^2} +\frac{2}{n^2\ln 2} \phi(n) + \mathcal{O}(n^{-2.99}), $$
where
$$ \phi(x) = \sum_{k\ne 0} \Gamma\left(\frac{2\pi i k}{\ln 2}\right)\zeta\left(\frac{2\pi i k}{\ln 2}\right)x^{-\frac{2\pi i k}{\ln 2}}. $$