On a conjecture of Erdös and certain Dirichlet series

Abstract

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 true. If $q\equiv 3$ ( mod $4$), Murty and Saradha proved the conjecture. We show that this conjecture is true for $82\%$ of the remaining integers $q\equiv 1$ ( mod $4$).

To appear in the Pacific Journal of Mathematics