On a conjecture of erdős and certain dirichlet series

Abstract

Let f : Z/qZ{long rightwards arrow}Z be such that f (a)=±1 for 1≤a <q, and f(q)=0. Then Erdős conjectured that Σn≥1 f(n)/n≠0. For q even, it is easy to show that the conjecture is true. The case q ≡ 3 (mod 4) was solved by Murty and Saradha. In this paper, we show that this conjecture is true for 82% of the remaining integers q ≡ 1 (mod 4).