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By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent.

  • Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{R}^+:\sum_{n\geq 1}\frac{\sin(n^2)}{n^\alpha}\mbox{ is convergent}\right\}?$$
  • In the case of a positive answer to the previous question, what is $$\inf\left\{\beta\in\mathbb{R}^+:\sum_{n\geq 1}\frac{\sin(n^2)}{\sqrt{n}(\log n)^\beta}\mbox{ is convergent}\right\}?$$
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By modelling $\sin(n^2)$ as a sequence of independent random variables $X_n$, I would expect a positive answer to the first question, also I would expect the later series to be convergent when $\beta > 1/2$ and divergent when $\beta < 1/2$. Thus the answer to the second question would be $1/2$. – blabler Nov 30 '12 at 4:13
    
A related question has popped up since this one was asked. – Douglas B. Staple Mar 29 '13 at 3:44
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Applying the argument here, math.stackexchange.com/questions/2270/… we obtain that the first quantity is $\leq \frac{7}{8}$ – i707107 Jun 8 '13 at 17:33

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