Vinogradov's Theorem/Major Arcs

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

Then:
 * $\displaystyle \int_\mathcal M \map F \alpha^3 \map e {-N \alpha} \rd \alpha = \frac {N^2} 2 \map {\mathcal S} N + \map {\mathcal O} {\frac {N^2} {\paren {\ln N}^{B/2} } }$

where the implied constant depends only on $B$.

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

Let $\beta \in \R$.

Let:
 * $\displaystyle \map u \beta = \sum_{n \mathop \le N} \map e {n \beta}$

For $P \ge 1$, define:


 * $\displaystyle \map {J_P} N = \int_{-P/N}^{P/N} \map u \beta^3 \map e {-N \beta} \rd \beta$
 * $\map J N = \map {J_{N/2} } N$

Then with $Q = \paren {\ln N}^B$ as above:


 * $\map {J_Q} N = \map J N + \map {\mathcal O} {\dfrac {N^2} {Q^2} }$

and:


 * $\map J N = \dfrac {N^2} 2 + \map {\mathcal O} N$

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

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

Then:


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

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