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Is the series $$\sum_{n\ge 1} \frac{\sin(n^2)}{n}$$ convergent?

My thoughts so far:

1) This is an alternating series so the integration test does not work here.

2) The Weyl inequality roughly says $$\sum_{n\le N} \sin(n^2)$$ is $O(N^{1/2+\epsilon})$, so the Dirichlet test does not work directly, but one can take $$a_n=n^{-1},b_n=\sum_{k\le n} \sin(k^2)$$ and follow the idea of Dirichlet test. The problem now is that the Weyl bound does not hold for all $N$.

Robert Z
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AlgRev
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  • See also http://math.stackexchange.com/questions/342637/does-sum-dfrac-sin-nn-converge. – Dietrich Burde Mar 27 '15 at 15:23
  • What are your thoughts so far? To get you started, notice that the sum of $(-1)^n/n$ converges while the sum of $1/n$ diverges. This sum is between them, so there is some question about how $\sin(n^2)$ behaves. If the result is true, then Dirichlet's test can probably prove it. – Ian Mar 27 '15 at 15:24
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    @Chou Thank you. I think sin(x^2) and sin(x) behave much differently. – AlgRev Mar 27 '15 at 15:49
  • Is there any known result, apart from Weyl's, about the estimates for the partial sums $\sum_{1}^{m}\sin n^{2}$? I would say you may instead make your question as seeking after the estimates for the sequence of the partial sums. – Yes Mar 27 '15 at 15:49
  • @Chou Good point. That is kind of what I am asking for. – AlgRev Mar 27 '15 at 15:50

1 Answers1

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You are on the right track. The key is to consider partial sums: $$ S_N = \sum_{n=1}^{N}\frac{\sin(n^2)}{n} $$ then find a good rational approximation of $\pi$ depending on $N$, apply Weyl bound (or Weyl differencing technique) to estimate $\sum_{n=1}^{k}e^{in^2}$ and finish through partial summation.

Details on page $11$ here (it is in Italian, hope you don't mind).

Jack D'Aurizio
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