Asymptotic formula for the moments of Takagi function

E. A. Timofeev

Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 1, pp. 5-11

Résumé

Takagi function is a simple example of a continuous but nowhere differentiable function. It is defined by $$ T(x) = \sum_{k=0}^{\infty}2^{-n}\rho(2^nx), $$ where $$ \rho(x) = \min_{k\in \mathbb{Z}}|x-k|. $$ The moments of Takagi function are defined as $$ M_n = \int_0^1\,x^n T(x)\,dx. $$ The main result of this paper is the following: $$ M_n = \frac{\ln n - \Gamma'(1)-\ln\pi}{n^2\ln 2}+\frac{1}{2n^2} +\frac{2}{n^2\ln 2} \phi(n) + \mathcal{O}(n^{-2.99}), $$ where $$ \phi(x) = \sum_{k\ne 0} \Gamma\left(\frac{2\pi i k}{\ln 2}\right)\zeta\left(\frac{2\pi i k}{\ln 2}\right)x^{-\frac{2\pi i k}{\ln 2}}. $$

Détail

Mots-clés : moments
Keywords: self-similar, Takagi function, singular, Mellin transform, asymptotic.