ALH-2018

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 $C=\{1\}$.