Asymptotic Formula for the Moments of the Takagi Function

Abstract

The Takagi function is a simple example of a continuous yet nowhere differentiable function and is given as T(x) = Σ_k=0^∞ 2⁻ⁿ ρ(2ⁿx), where \(\rho (x) = \mathop {\min }\limits_{k \in \mathbb{Z}} |x - k|\). The moments of the Takagi function are given as M_n = ∫₀¹xⁿT(x)dx. The estimate \({M_n} = \frac{{1nn - \Gamma '(1) - 1n\pi }}{{{n^2}1n2}} + \frac{1}{{2{n^2}}} + \frac{2}{{{n^2}1n2}}\phi (n) + O({n^{ - 2.99}})\), where the function \(\phi (x) = \sum\nolimits_{k \ne 0} \Gamma (\frac{{2\pi ik}}{{1n2}})\zeta (\frac{{2\pi ik}}{{1n2}}){x^{ - \frac{{2\pi ik}}{{1n2}}}}\) is periodic in log₂x and Γ(x) and ζ(x) denote the gamma and zeta functions, is the principal result of this work.