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Fix $\alpha \notin \mathbb{Q}$. I would like to know a reference that shows $$\mathbb{E}_{n\leq N} \Lambda(n) e^{2\pi i \alpha n} \to 0,$$ as $N$ tends to infinity.

I am familiar with Vinagradov's estimate, which seemingly does not directly imply this.

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    $\begingroup$ Two good references for this result are Chapter 13 of Iwaniec and Kowalski's book "Analytic Number Theory," (Thm. 13.6) and recent course notes by Tao, terrytao.wordpress.com/2015/03/30/… (combine Prop 18 and 24). $\endgroup$
    – Brad Rodgers
    Jan 7, 2016 at 2:00
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    $\begingroup$ @BradRodgers Many thanks for your response. Proposition 24 in Tao's notes was exactly what I was looking for: since $\alpha$ is irrational, one can assume that $q \geq C$ for any $C \geq 1$ and get savings from the $\phi(q)$. In Iwaniec/Kowalski, there is an annoying $log(x)^3$ which prevents one from obtaining the result in the case where $\alpha$ is close to a rational with tiny denominator infinitely often (perhaps something like the Louisville number). $\endgroup$
    – George Shakan
    Jan 7, 2016 at 17:03
  • $\begingroup$ This gives a nice major arc (Proposition 24)/minor arc (Vinagradov's estimate proof). Also in Tao's notes, he mentions that in Helfgott's proof of ternary Goldbach he eliminates the extra logs from Proposition 18, which seems very nontrivial. $\endgroup$
    – George Shakan
    Jan 7, 2016 at 17:10
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    $\begingroup$ Ah, I didn't pay sufficient attention to the $(\log x)^3$, but you're absolutely right. $\endgroup$
    – Brad Rodgers
    Jan 8, 2016 at 3:20

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