Erdős's nonvanishing conjecture for periodic sign functions

Let $N$ be a positive integer, and let $f$ be an arithmetic function with period $N$ such that $f(n)\in\{-1,1\}$ for $n=1,2,\dots,N-1$ and $f(n)=0$ whenever $n\equiv 0\pmod{N}$. Erdős's conjecture. One has

$$\sum_{n\ge 1}\frac{f(n)}{n}\neq 0.$$

This conjecture concerns the nonvanishing of the associated Dirichlet series at $1$ for periodic sign-valued functions. The source attributes it to Erdős and describes the paper as proving the conjecture for a new family of natural numbers; its general status is not specified here.