Erdős Problem #318 — Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function.

Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite non-empty $S\subset A$ such that $\sum_{n\in S}\frac{f(n)}{n}=0?$ What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$?