Polylogarithms and the Asymptotic Formula for the Moments of Lebesgue’s Singular Function

Abstract

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
$$
\tau(x) =
\frac12\ln p + \Gamma'(1)\log_2 p +\frac1{\ln 2}\frac{\partial}{\partial z}\left.Li_{z}\left(-\frac{q}{p}\right)\right|_{z=1}
%+\\
\\
+\frac1{\ln 2}\sum_{k\ne0}
\Gamma(z_k)Li_{z_k+1}\left(-\frac{q}{p}\right) x^{-z_k},
$$
$$
z_k = \frac{2\pi ik}{\ln 2}, \ \ k\ne 0.
$$
The proof is based on analytic techniques such as the poissonization and the Mellin transform.