# Arithmetic of the Fabius function

## TL;DR

This paper proves that certain scaled values of the Fabius function at dyadic points are integers and explores their arithmetical properties, unifying various notations and extending understanding of this classical function.

## Contribution

The paper establishes the integrality of specific scaled Fabius function values at dyadic points and clarifies notation, advancing the arithmetic understanding of the function.

## Key findings

- The numbers R_n are integers.
- Identifies arithmetical properties of Fabius function at dyadic points.
- Unifies notations related to the Fabius function.

## 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.