  Let $f:\Z/q\Z\rightarrow\Z$ be such that $f(a)=\pm 1$ for $1\le a<q$, and
$f(q)=0$. Then Erd\"os conjectured that $\sum_{n\ge1}\frac{f(n)}{n} \ne 0$. 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 $4$).
