Vinogradov's Theorem/Major Arcs

Theorem
For any $B > 0$,


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

where the implied constant depends only on $B$.

Lemma 2
For $N \ge 1$, $\beta \in \R$, let $u(\beta) = \sum_{n \le N}e(n\beta)$. For $P \ge 1$, define:


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

Then with $Q = (\log N)^B$ as above,


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

and


 * $\displaystyle J(N) = \frac {N^2}2 + \mathcal O(N)$

Lemma 3
Let $\alpha \in \mathcal M(q,a)$ for some $q,a$ such that $\mathcal M(q,a) \subseteq \mathcal M$, and let $\beta = \alpha - \displaystyle frac aq$.

Then:


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

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