Vinogradov's Theorem/Corollary 1

Corollary to Vinogradov's Theorem

Let $\SS$ be the arithmetic function:

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

where:

$p$ ranges over the primes

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

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

Let:

$\ds \map r N = \sum_{p_1 + p_2 + p_3 \mathop = N} 1$

where $p_1, p_2, p_3$ are prime.

Then for sufficiently large odd integers $N$:

$\map r N = \map \SS N \dfrac {N^2} {2 \paren {\log N}^3} \paren {1 + \map \OO {\dfrac {\log \log N} {\log N} } }$

where $\OO$ denotes big-O notation.

Proof

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