Vinogradov's Theorem/Major Arcs

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Theorem

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

Then:

$\ds \int_\MM \map F \alpha^3 \map e {-N \alpha} \rd \alpha = \frac {N^2} 2 \map \SS N + \map \OO {\frac {N^2} {\paren {\ln N}^{B/2} } }$

where the implied constant depends only on $B$.

Proof

Lemma 1

Let $\phi$ be the Euler $\phi$ function.

Let $\mu$ be the Möbius function.

Let $c_q$ be the Ramanujan sum modulo $q$.

Let $P, N \ge 1$.

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Let:

$\ds \map {\SS_P} N := \sum_{q \mathop \le P} \frac {\map \mu q \map {c_q} N} {\map \phi q^3}$

$\ds \map \SS N := \lim_{P \mathop \to \infty} \map {\SS_P} N$

Then:

$\map \SS N = \map {\SS_P} N + \map \OO {P^{\epsilon -1} }$

and $\SS$ has the Euler product:

$\ds \map \SS N = \prod_{p \mathop \nmid N} \paren {1 + \frac 1 {\paren {p - 1}^3} } \prod_{p \mathop \divides N} \paren {1 - \frac 1 {\paren {p - 1}^2} }$

where:

$p \nmid N$ denotes that $p$ is not a divisor of $N$

$p \divides N$ denotes that $p$ is a divisor of $N$.

$\Box$

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Lemma 2

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

Let $\beta \in \R$.

Let:

$\ds \map u \beta = \sum_{n \mathop \le N} \map e {n \beta}$

For $P \ge 1$, define:

$\ds \map {J_P} N = \int_{-P/N}^{P/N} \map u \beta^3 \map e {-N \beta} \rd \beta$

$\map J N = \map {J_{N/2} } N$

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

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$\map {J_Q} N = \map J N + \map \OO {\dfrac {N^2} {Q^2} }$

and:

$\map J N = \dfrac {N^2} 2 + \map \OO N$

$\Box$

Lemma 3

Let $\alpha \in \map \MM {q, a}$ for some $q, a$ such that:

$\map \MM {q, a} \subseteq \MM$

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

Then:

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

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

$\Box$

$\blacksquare$