For which p-adic integers x can Sum_k binomial(x,k)^(-1) be defined?

TL;DR

This paper investigates the p-adic definability of the inverse binomial sum function, revealing its discontinuity, specific definability conditions for p-adic integers, and exceptional cases for small primes.

Contribution

It characterizes when the inverse binomial sum function is p-definable for p-adic integers, including new results for all primes and special cases for small primes.

Findings

f(-1) is p-definable for all primes p

For odd p, -1 is the only non-natural p-adic integer with definability

For p=2, definability depends on binary expansion sparsity

Abstract

Let f(n)= Sum binomial(n,k)^(-1). First, we show that f:N to Q_p is nowhere continuous in the p-adic topology. If x is a p-adic integer, we say that f(x) is p-definable if lim f(x_j) exists in Q_p, where x_j denotes the jth partial sum for x. We prove that f(-1) is p-definable for all primes p, and 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 show that if k is a positive integer, then f(-k-1) is not 2-definable, but that if the 1's in the binary expansion of x are eventually very sparse, then f(x) is 2-definable. Some of our proofs require that p satisfy one of two conditions. There are three small primes that do not satisfy the relevant condition, but our theorems can be proved directly for these primes. No other prime less than 100,000,000 fails to satisfy the conditions.