On the rational approximation of the sum of the reciprocals of the Fermat numbers

TL;DR

This paper investigates the irrationality measures of special functions related to Fermat numbers, proving they have irrationality exponent exactly 2, and establishes non-vanishing determinants of certain Hankel matrices associated with these functions.

Contribution

It proves the non-vanishing of Hankel determinants for specific sequences and determines the irrationality exponent of the sum of reciprocals of Fermat numbers as exactly 2.

Findings

Determinants of certain Hankel matrices are nonzero for all sizes.

Irrationality exponents of the functions at reciprocal integers are exactly 2.

Sum of reciprocals of Fermat numbers has irrationality exponent 2.

Abstract

Let $\C{G}(z):=\sum_{n=0}^\infty z^{2^n}(1-z^{2^n})^{-1}$ denote the generating function of the ruler function, and $\C{F}(z):=\sum_{n=0}^\infty z^{2^n}(1+z^{2^n})^{-1}$; note that the special value $\C{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_n:=2^{2^n}+1$. The functions $\C{F}(z)$ and $\C{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\C{F}(\ga)$ and $\C{G}(\ga)$ are transcendental for all algebraic numbers $\ga$ which satisfy $0<\ga<1$. For a sequence $\mathbf{u}$, denote the Hankel matrix $H_n^p(\mathbf{u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$. Let $\ga$ be a real number. The {\em irrationality exponent} $\mu(\ga)$ is defined as the supremum of the set of real numbers $\mu$ such that the inequality $|\ga-p/q|<q^{-\mu}$ has infinitely many solutions $(p,q)\in\B{Z}\times\B{N}.$ In this paper, we first prove that the determinants of $H_n^1(\mathbf{g})$ and $H_n^1(\mathbf{f})$ are nonzero for every $n\geqslant 1$. We then use this result to prove that for $b\geqslant 2$ the irrationality exponents $\mu(\C{F}(1/b))$ and $\mu(\C{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.