The behavior of $\sum_{n=1}^\infty\zeta^{\integerpart{n\theta}}/n$ for particular values of~$\theta$

Let $\zeta$ be a primitive $q''$-root of unity. We prove that the series $\sum_{n=1}^\infty \zeta^{\integerpart{n\theta}}/n$ for $\theta\in\Q$ converges if and only if $\theta=p/q$ with $(p,q)=1$ and $q''\nmid p$, and that there exists an uncountable set $\Set$ of Liouville''s numbers such that the series does not converge when $\theta\in\Set$.