Vinogradov's Theorem/Minor Arcs

Theorem
Let $\ds \map F \alpha = \sum_{n \mathop \le N} \map \Lambda n \map e {\alpha n}$. For any $B > 0$:
 * $\ds \int_\MM \map F \alpha^3 \map e {-\alpha N} \rd \alpha \ll \frac {N^2} {\paren {\ln N}^{B/2 - 5} }$

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


 * $(1): \quad \ds \size {\alpha - \frac a q} \le \frac 1 {q^2} \quad 1 \le q \le N, \quad \gcd \set {a, q} = 1$

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

Then:
 * $\ds \sum_{k \mathop = 1}^m \min \set {\frac {m n} k, \frac 1 {\norm {\alpha k} } } \ll \paren {m + \frac {m n} q + q} \map \ln {2 q m}$

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 $\size {\beta_k}, \size {\gamma_k} \le 1$.

Then:


 * $\ds \sum_{k \mathop = y}^{x/y} \sum_{\ell \mathop = y}^{x/k} \alpha_k \beta_\ell \map e {\alpha k \ell} \ll x^{1/2} \paren {\ln x}^2 \paren {\frac x y + y + \frac x q + q}^{1/2}$