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

E. A. Timofeev

Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 5, pp. 595-602

Résumé

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.

Détail

Mots-clés : moments, Lebesgue’s function, polylogarithm
Keywords: self-similar, singular, Mellin transform, asymptotic.