I was investigating series whose convergence depends on a number's irrationality measure and would like to know if the sorts of questions I ask have been considered before and where I could look to find out more.
Suppose $x$ is an irrational number. It is easy to prove that the series $$ \sum_{n=1}^{\infty} \frac{\csc(\pi nx)}{n^s} $$ converges absolutely whenever $s>\mu(x)$, where $\mu(x)$ is the irrationality measure of $x$. Now let's suppose we set $x=L$, where $L = \sum_{n=1}^{\infty} \frac{1}{10^{n!}} $ is Liouville's constant. It follows that the above series will diverge for any $s$, but what if we modify things by dividing by a factorial instead of a power function? Does the series $$ \sum_{n=1}^{\infty} \frac{\csc(\pi n L)}{n!} $$ converge? I suppose this question really comes down to knowing the exact growth function $\psi$ where $|L-\frac{p}{q}|<\psi(q)$ will have finitely many solutions but infinitely many for a function growing just a bit slower. Do we know the exact growth function?
My follow up question: Does there exist a function $f$ which grows so fast that the series $$ \sum_{n=1}^{\infty} \frac{\csc(\pi nx)}{f(n)} $$ converges for any irrational $x$, or will there always be some small set of very well approximable irrationals for which it will diverge?
Thanks for any help.