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 $\theta\in\Set$.
The behavior of $\sum_{n=1}^\infty\zeta^{\integerpart{n\theta}}/n$ for particular values of~$\theta$ / G. Molteni. - In: ACTA MATHEMATICA HUNGARICA. - ISSN 0236-5294. - 117:1-2(2007), pp. 61-76.
The behavior of $\sum_{n=1}^\infty\zeta^{\integerpart{n\theta}}/n$ for particular values of~$\theta$
G. Molteni
2007
Abstract
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 $\theta\in\Set$.
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