Abstract:
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
$$
\phi(x) =
\sum_{k\ne 0}
\Gamma\left(\frac{2\pi i k}{\ln 2}\right)\zeta\left(\frac{2\pi i k}{\ln 2}\right)x^{-\frac{2\pi i k}{\ln 2}}.
$$
Fulltext:
PDF file (540 kB)