For which 2-adic integers $x$ can $\sum_k \binom xk^{-1}$ be defined?

TL;DR

This paper investigates conditions under which the sum of inverse binomial coefficients, defined for 2-adic integers, is well-defined in the 2-adic number system, extending previous results and proposing new conjectures.

Contribution

It introduces new conjectures suggesting broader classes of 2-adic integers for which the sum is 2-definable, building on prior work that characterized such integers for odd primes.

Findings

For odd p, only -1 in Z_p-N is p-definable.

For p=2, sparsity in binary expansion implies 2-definability.

New conjectures propose more 2-adic integers with definable sums.

Abstract

Let $f(n)=\sum_k \binom nk^{-1}$. 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.