The behavior of \(\sum_{n=1}^\infty \zeta^{\lfloor n\theta\rfloor}/n\) for particular values of \(\theta\) (Q955170)

Let \(\zeta\) be a primitive \(Q\)-root of unity and \(\eta\in\mathbb Q\). The author proves that the series \(\sum_{n=1}^\infty \zeta^{[n\eta]}\) converges iff \(\eta=\frac pq\) with \((p,q)=1\) and \(Q\) does not divide \(p\). The fact that there exists uncountable set \(\mathbb S\) of Liouville numbers such that the series does not converge for \(\eta\in \mathbb S\) is also included. The proofs are based on the exact calculations of partial sums of the above series and use congruences.