  For a rational valued periodic function, we associate a Dirichlet series and
provide a new necessary and sufficient condition for the vanishing of this
Dirichlet series specialized at positive integers. This question was initiated
by Chowla, and carried out by Okada for a particular infinite sum. Our approach
relies on the decomposition of the Dirichlet characters in terms of primitive
characters. Using this, we find some new family of natural numbers for which a
conjecture of Erd\"os holds. We also characterize rational valued periodic
functions for which the associated Dirichlet series vanishes at two different
positive integers under some additional conditions.
