Vinogradov's Theorem/Major Arcs

Theorem
Let $B \in \R_{>0}$.

Then:
 * $\displaystyle \int_\mathcal M F \left({\alpha}\right)^3 e \left({-N \alpha}\right) \, \mathrm d \alpha = \frac {N^2} 2 \mathcal S \left({N}\right) + \mathcal O \left({\frac {N^2} {\left({\ln N}\right)^{B/2} } }\right)$

where the implied constant depends only on $B$.

Lemma 2
Let $N \in \N_{\ge 1}$.

Let $\beta \in \R$.

Let:
 * $\displaystyle u \left({\beta}\right) = \sum_{n \mathop \le N} e \left({n \beta}\right)$

For $P \ge 1$, define:


 * $\displaystyle J_P \left({N}\right) = \int_{-P/N}^{P/N} u \left({\beta}\right)^3 e \left({-N \beta}\right) \, \mathrm d \beta$
 * $J \left({N}\right) = J_{N/2} \left({N}\right)$

Then with $Q = \left({\ln N}\right)^B$ as above:


 * $J_Q \left({N}\right) = J \left({N}\right) + \mathcal O \left( \dfrac {N^2} {Q^2} \right)$

and:


 * $J \left({N}\right) = \dfrac {N^2} 2 + \mathcal O \left({N}\right)$

Lemma 3
Let $\alpha \in \mathcal M \left({q, a}\right)$ for some $q, a$ such that:
 * $\mathcal M \left({q, a}\right) \subseteq \mathcal M$

Let $\beta = \alpha - \dfrac a q$.

Then:


 * $F \left({\alpha}\right)^3 = \dfrac {\mu \left({q}\right)} {\phi \left({q}\right)^3} u \left({\beta}\right)^3 + \mathcal O \left({N^3 \exp \left({-C \sqrt {\ln N} }\right)}\right)$

where $C$ is a constant that depends only on $B$.