Vinogradov's Theorem/Minor Arcs

Theorem
Let $\displaystyle F \left({\alpha}\right) = \sum_{n \mathop \le N} \Lambda \left({n}\right) e \left({\alpha n}\right)$. For any $B > 0$:
 * $\displaystyle \int_\mathcal M F \left({\alpha}\right)^3 e \left({-\alpha N}\right) \, \mathrm d \alpha \ll \frac {N^2} {\left({\ln N}\right)^{B/2 - 5} }$

Lemma 2
Let $a, q \in \Z$ such that:


 * $(1): \quad \displaystyle \left\vert{\alpha - \frac a q }\right\vert \le \frac 1 {q^2} \quad 1 \le q \le N, \quad \gcd \left({a, q}\right) = 1$

Let $m, n \in \Z$ be any integers.

Then:
 * $\displaystyle\sum_{k mathop = 1}^m \min \left\{ {\frac {m n} k, \frac 1 {\left\Vert{\alpha k}\right\Vert} }\right\} \ll \left({m + \frac {m n} q + q}\right) \ln \left({2 q m}\right)$

Lemma 3
Let $\alpha$ satisfy the condition $(1)$ of Lemma 2.

Let $x, y \in \N$.

Let $\beta_k, \gamma_k$ be any complex numbers such that $\left\vert{\beta_k}\right\vert, \left\vert{\gamma_k}\right\vert \le 1$.

Then:


 * $\displaystyle \sum_{k \mathop = y}^{x/y} \sum_{\ell \mathop = y}^{x/k} \alpha_k \beta_\ell e \left({\alpha k \ell}\right) \ll x^{1/2} \left({\ln x}\right)^2 \left({\frac x y + y + \frac x q + q} \right)^{1/2}$