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

• Main Author: E. A. Timofeev
• Format: Article
• Language: English
• Published: Yaroslavl State University 2016-10-01
• Series: Modelirovanie i Analiz Informacionnyh Sistem
• Subjects: moments; self-similar; lebesgue’s function; singular; mellin transform; polylogarithm; asymptotic
• Online Access: https://www.mais-journal.ru/jour/article/view/393

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.