For which 2-adic integers $x$ can ${\sum }_k{\left(\genfrac{}{}{0ex}{}{x}{k}\right)}^{-1}$ be defined?

Abstract

Let $f(n)={\sum }_k{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^{-1}$. In a previous paper, we defined for a p-adic integer x that f(x) is p-definable if lim $f(x_j)$ exists in $Q_p$, where $x_j$ denotes the mod $p^j$ reduction of $x$. We proved that if p is odd, then -1 is the only element of $Z_p-N$ for which f(x) is p-definable. For p=2, we proved that if the 1's in the binary expansion of x are eventually extraordinarily sparse, then f(x) is 2-definable. Here we present some conjectures that f(x) is 2-definable for many more 2-adic integers. We discuss the extent to which we can prove these conjectures.

Preliminary report