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Consider the following series for $x\notin \mathbb{Q}$:

$$f(x)=\sum_{n=1}^{\infty}\frac{1}{n^3\sin(n\pi x)}$$

When does it converge? By Khintchin theorem, I know that it converges almost surely but can we say more?

For the curious reader, this sum (restricted to odd values of $n$) actually appears in the asymptotic behavior of signatures in TQFT.

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    $\begingroup$ this stuff is related to the irrationality measure of $x$ - it is easy to construct numbers for which $|x-p_m/q_m| << q_m^{-4}$ for example for infinitely many $m$ and some (distinct) fractions $p_m/q_m$ and then $|\pi q_mx-\pi p_m| << q_m^{-3}$ so $1/q_m^3\sin q_m \pi x$ doesn't converge to $0$ and the series perforce diverges etc; for $x=1/\pi$ this is related to the well known Flint Hills problem and the irrationality measure of $\pi$ $\endgroup$
    – Conrad
    Commented Nov 12 at 16:26
  • $\begingroup$ Thanks Conrad for your answer, I did not know about the Flint Hills problem. Any other comments on the properties of this function are welcome. $\endgroup$
    – Julien Marché
    Commented Nov 13 at 9:29

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