Asymptotic Formula for the Moments of the Takagi Function
- Autores: Timofeev E.1
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Afiliações:
- Demidov Yaroslavl State University
- Edição: Volume 51, Nº 7 (2017)
- Páginas: 731-735
- Seção: Article
- URL: https://journals.rcsi.science/0146-4116/article/view/175380
- DOI: https://doi.org/10.3103/S0146411617070197
- ID: 175380
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Resumo
The Takagi function is a simple example of a continuous yet nowhere differentiable function and is given as T(x) = Σk=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 Mn = ∫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.
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Sobre autores
E. Timofeev
Demidov Yaroslavl State University
Autor responsável pela correspondência
Email: timofeevea@gmail.com
Rússia, Yaroslavl, 150003