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Arithmetic of the Fabius function

Name: Arithmetic of the Fabius function
Creator: Arias de Reyna, J.
Keywords: Mathematics - Number Theory, FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, 01 natural sciences, 11B83 (Primary), 11B68 (Secondary)

descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 01 Jan 2017Embargo end date: 01 Jan 2017Publisher:arXiv

Authors: Arias de Reyna, J.;

doi: 10.48550/arxiv.1702.06487 , 10.5281/zenodo.10677753

arXiv: http://arxiv.org/abs/1702.06487

Arithmetic of the Fabius function

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Abstract

I solve here a question of Vladimir Reshetnikov in Mathoverflow (question 261649) about the values of Fabius function. Namely, I prove that the numbers $R_n:=2^{-\binom{n-1}{2}}(2n)! F(2^{-n})\prod_{m=1}^{\lfloor n/2\rfloor}(2^{2m}-1)$ are integers. We show also some other arithmetical properties of the values of Fabius function at dyadic points. The Fabius function was defined in 1935 by Jessen and Wintner and has been independently defined at least six times since. We attempt to unify notations related to the Fabius function.

15 pages

Keywords

Mathematics - Number Theory, FOS: Mathematics, Number Theory (math.NT), 11B83 (Primary), 11B68 (Secondary)

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