In a written communication to Livingston, Paul Erdős proposed the following conjecture:

If $N$ is a positive integer and $f$ is an arithmetic function with period $N$ and $f(n) \in \{-1,1\}$ when $n= 1,2,\dots ,N-1$ and $f(n)=0 $ whenever $n\equiv 0 \bmod{~} N$ then $\displaystyle \sum \limits_{n \ge 1} \frac{f(n)}{n} \neq 0$.

We describe the literature around this conjecture, and mention some new results. This is an ongoing joint work with Ram Murty and Siddhi Pathak.