Title. Sums of finitely many distinct reciprocals
Speaker. Sylvia Silberger
Institution. Hofstra University, Hemstead, NY
Abstract. Let $$\mathcal F$$ denote the family of all finite nonempty $$S\subseteq{\mathbb N}:={1,2,\ldots}$$, and let $$\mathcal F(X):=\mathcal F \cap {S:S\subseteq X}$$ when $$X\subseteq{\mathbb N}$$.
In this talk we treat the function $$\sigma:{\cal F}\rightarrow{\mathbb Q}^+$$ given by $$\sigma:S\mapsto\sigma S :=\sum{1/x:x\in S}$$, and the function $$\delta:{\cal F}\rightarrow{\mathbb N}$$
defined by $$\sigma S = \nu S/\delta S$$ where the integers $$\nu S$$ and $$\delta S$$ are coprime.
We then discuss the following results.
Theorem 1.1.
For each $$r\in{\mathbb Q}^+$$, there exists an infinite pairwise disjoint subfamily $${\cal H}_r\subseteq{\cal F}$$ such that $$r=\sigma S$$ for all $$S\in{\cal H}_r$$.
Theorem 1.2.
Let $$X$$ be a pairwise coprime set of positive integers. Then $$\sigma\upharpoonright \mathcal F (X)$$ and $$\delta\upharpoonright \mathcal F (X)$$ are injective. Also, $$\sigma C\in{\mathbb N}$$ for
$$C\in{\cal F}(X)$$ only if $$C={1}$$.