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Let $$F_n(\alpha) = \prod_{k = 1}^n \Phi_k(e(\alpha))$$ where $e(\alpha) = e^{2\pi i\alpha}$

It is clear that $F_n(\alpha) = 0$ iff $\alpha = \frac{a}{q}$ for relatively prime $a, q$ s.t. $q \le n$.

Now, let $\frac{p_1}{q_1}, \frac{p_2}{q_2}$ be two such rationals s.t there are no such rationals between the two. How would one estimate values of $F_n(\alpha)$ if $\frac{p_1}{q_1} < \alpha < \frac{p_2}{q_2}$.

Mayank Pandey
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    I believe this is a key part of using the Hardy-Littlewood Circle Method. – Gerry Myerson Jan 13 '14 at 03:27
  • Do you mean that this can be solved with the Hardy-Littlewood circle method? – Mayank Pandey Jan 13 '14 at 03:50
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    The method breaks the circle up into arcs near a rational with small denominator ("major arcs") and the rest ("minor arcs") and uses different methods to make estimates in the two cases. You are trying to make an estimate on the minor arcs, so you may find something useful in a discussion of the method. – Gerry Myerson Jan 13 '14 at 13:28
  • I think it is useful to remark that if $\frac{p_1}{q_1}$ and $\frac{p_2}{q_2}$ are two consecutive terms of the same Farey sequence, then their absolute difference is exactly $\frac{1}{q_1 q_2}$, so you "only" need to estimate the derivative of $F_n$ in its roots $\frac{p_1}{q_1},\frac{p_2}{q_2}$. – Jack D'Aurizio Jan 29 '14 at 10:32

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