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Suppose we have $\alpha \in \mathbb{R}$ such that $|\alpha - a/q| < 1/q^2$, where $(a,q)=1$. Then we know that the exponential sum $$ S(\alpha) = \sum_{1 \leq n \leq X} \Lambda(n) e(n \alpha) $$ can be bounded by using Vaughn's identity, and for example, it is in Multiplicative Number Theory by Davenport. I was wondering if the sum of the following type can be estimated in a similar manner, for $w \in \mathbb{Z}$, $$ T_w(\alpha) = \sum_{1 \leq n \leq X} \Lambda(n) \Lambda (n + w) e(n \alpha), $$ or by different method.

I was wondering if there are any references for sum of this type? I would also appreciate any comments/answers.

Thank you very much!

PS $\Lambda$ is the von-Mangoldt function, $e(m) = e^{2 \pi i m}$ here.

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  • $\begingroup$ AFAIK, such estimates are out of the reach of current methods, for the specific case when $\alpha=0$ and $w=2$ is equivalent to the twin prime conjecture. $\endgroup$ – Fan Zheng Feb 18 '16 at 14:58

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