Polylogarithms and the asymptotic formula for the moments of Lebesgue’s singular function
E. A. Timofeev P.G. Demidov Yaroslavl State University,
14 Sovetskaya str., Yaroslavl 150003, Russia
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
\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.
Keywords:
moments, self-similar, Lebesgue’s function, singular, Mellin transform, polylogarithm, asymptotic.
UDC:
519.17 Received: 10.07.2016
DOI:
10.18255/1818-1015-2016-5-595-602
Fulltext:
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