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

https://doi.org/10.18255/1818-1015-2016-5-595-602

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.

About the Author

E. A. Timofeev
P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia
Russian Federation
ScD, professor


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Review

For citations:


Timofeev E.A. Polylogarithms and the Asymptotic Formula for the Moments of Lebesgue’s Singular Function. Modeling and Analysis of Information Systems. 2016;23(5):595-602. (In Russ.) https://doi.org/10.18255/1818-1015-2016-5-595-602

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ISSN 1818-1015 (Print)
ISSN 2313-5417 (Online)