Problem 318

Erdős Problem #318: declared status 'solved'. Formalized: yes. 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 $1$? Current best: For the squares $1$ must be excluded or the result is trivially false, since$$\sum_{k\geq 2}\frac{1}{k^2}<1.$$This is false for some sets $A$ of positive density - indeed, it fails for any set $A$ containing exactly one even number. Prize: no. Tags: number theory, unit fractions.