Vinogradov's Theorem/Major Arcs

Theorem

For any $B > 0$,

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

where the implied constant depends only on $B$.

Proof

Lemma 1

Let $\phi$ be the Euler phi function, $\mu$ the Mobius function and $c_q$ the Ramanujan sum modulo $q$.

For $P,N \geq 1$, define:

$\displaystyle \mathfrak S_P(N) = \sum_{q \leq P}\frac{\mu(q)c_q(N)}{\phi(q)^3},\quad \mathfrak S(N) = \lim_{P \to \infty} \mathfrak S_P(N)$

Then $\mathfrak S(N) = \mathfrak S_P(N) + \mathcal O(P^{\epsilon -1})$ and $\mathfrak S$ has the Euler product:

$\displaystyle\mathfrak S(N) = \prod_{p \nmid N} \left( 1 + \frac 1{(p-1)^3} \right)\prod_{p \mid N} \left( 1 - \frac 1{(p-1)^2} \right)$

$\Box$

Lemma 2

For $N \geq 1$, $\beta \in \R$, let $u(\beta) = \sum_{n \leq N}e(n\beta)$. For $P \geq 1$, 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

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

$\Box$

Lemma 3

Let $\alpha \in \mathfrak M(q,a)$ for some $q,a$ such that $\mathfrak M(q,a) \subseteq \mathfrak M$, 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 $B$.

$\Box$

$\blacksquare$