Vinogradov's Theorem/Minor Arcs

Theorem

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Let $\ds \map F \alpha = \sum_{n \mathop \le N} \map \Lambda n \map e {\alpha n}$.

For any $B > 0$:

$\ds \int_\MM \map F \alpha^3 \map e {-\alpha N} \rd \alpha \ll \frac {N^2} {\paren {\ln N}^{B/2 - 5} }$

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Proof

Lemma 1

For $\beta \in \R$, define:

$\norm \beta := \min \set {\size {n - \beta}: n \in \Z}$

Then:

$\ds \forall \alpha \in \R: \size {\sum_{k \mathop = N_1}^{N_2} \map e {\alpha k} } \le \min \set {N_2 - N_1, \frac 1 {2 \norm \alpha} }$

$\Box$

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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:

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

$\Box$

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:

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

$\Box$

$\blacksquare$