On a conjecture of Erdös and certain Dirichlet series

Tapas Chatterjee, M. Ram Murty

Let $f:\mathbb{Z}/q\mathbb{Z}\to \mathbb{Z}$ be such that $f(a)=\pm 1$ for $1\le a<q$, and $f(q)=0$. Then Erdös conjectured that ${\sum }_{n\ge 1}\frac{f(n)}{n}\ne 0$. For $q$ even, this is trivially…