In this book on page 82 I found an estimate $$\sum_{j=1}^{n-2}\frac{\sin(k+1)\theta_{j}}{\sin\theta_{j}}=O_{\epsilon}\left( p^{k\epsilon}\right)$$ as $k$ goes to infinity, for all $\epsilon>0$. Here $p$ is a prime number and $\theta_{1},...,\theta_{n-2}$ are complex numbers. After this estimate, the authors says: The last clearly implies that $\theta_{j}$ $\left(1\leq\theta_{j}\leq n-2\right)$ are real numbers. Why is this clearly true?

P.S. I hope this edited question is now ok.

  • $\begingroup$ Nothing is «clearly» true. $\endgroup$ – Mariano Suárez-Álvarez Jan 2 '14 at 19:17
  • $\begingroup$ I can't understand that part. So, if someone is clear how the author got the result, I would appreciate if you could write it. $\endgroup$ – Alem Jan 2 '14 at 19:26
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    $\begingroup$ Keep in mind that $|\sin z|$ can get large if $z =x+iy$ is complex (as large as $e^{|y|}$). So if any of the $\theta_j$ are complex, you can find values of $k$ for which the LHS grows exponentially in $k$ (this is obvious if there is a single $\theta_j$ whose imaginary part is largest, and you'll need some argument if there are several with the same largest imaginary part). Hope that helps. $\endgroup$ – Lucia Jan 2 '14 at 19:29
  • $\begingroup$ I really can't see the details. $\endgroup$ – Alem Jan 2 '14 at 19:56
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    $\begingroup$ @Alem In the future, when you post questions, please try to make the title informative. I suspect that the title that you used is a reason that people have voted to close your question. $\endgroup$ – Joe Silverman Jan 3 '14 at 0:01

A much more detailed writeup of this is is Davidoff/Sarnak/Vallette, see page 127 and on, especially p. 130.

  • $\begingroup$ @AlainValette I will take you up on it :) $\endgroup$ – Igor Rivin Jan 3 '14 at 20:37

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