Vinogradov's Theorem

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Theorem

Let $\Lambda$ be the von Mangoldt function.

For $N \in \Z$, let:

$\ds \map R N = \sum_{n_1 + n_2 + n_3 \mathop = N} \map \Lambda {n_1} \, \map \Lambda {n_2} \, \map \Lambda {n_3}$

be a weighted count of the number of representations of $N$ as a sum of three prime powers.

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 $N$.


Then for any $A > 0$ and sufficiently large odd integers $N$:

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

where $\OO$ denotes big-$\OO$ notation.


Corollary 1

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.


Corollary 2

Every sufficiently large odd integer is the sum of three prime numbers.


Proof

Throughout the proof, for $\alpha \in \R$, let the following notation be understood:

$\map e \alpha := \map \exp {2 \pi i \alpha}$

Let $B > 0$, and set $Q = \paren {\log N}^B$.

For $1 \le q \le Q, 0 \le a \le q$ such that $\gcd \set {a, q} = 1$, let:

$\map \MM {q, a} := \set {\alpha \in \closedint 0 1: \size {\alpha - \dfrac a q} \le \dfrac Q N}$

Let:

$\ds \MM := \bigcup {1 \mathop \le q \mathop \le Q} \bigcup_{\substack {0 \mathop \le a \mathop \le q} {\gcd \set {a, q} \mathop = 1} } \map \MM {q, a}$

be referred to as the major arcs.

Let:

$m := \closedint 0 1 \setminus \MM$

be referred to as the minor arcs.


Lemma 1

For sufficiently large $N$ the major arcs are pairwise disjoint, and the minor arcs are non-empty.

$\Box$


By the Vinogradov Circle Method (with $\ell = 3$ and $\AA$ the set of primes), letting $\ds \map F \alpha = \sum_{n \mathop \le N} \map \Lambda n \, \map e {\alpha n}$ we have:

$\ds \map R N = \int_0^1 \map F \alpha^3 \, \map e {-N \alpha} \rd \alpha$


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So by splitting the closed unit interval into a disjoint union:

$\closedint 0 1 = m \cup \MM$

we have:

$\ds \map R N = \int_m \map F \alpha^3 \, \map e {-\alpha N} \rd \alpha + \int_\MM \map F \alpha^3 \, \map e {-\alpha N} \rd \alpha$

We consider each of these integrals in turn.


Sum Over the Minor Arcs

For any $B > 0$:

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

$\Box$


Sum Over the Major Arcs

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 $B$.

$\Box$


Putting these estimates together, we obtain:

$\map R N = \dfrac {N^2} 2 \map \SS N + \map \OO {\dfrac {N^2} {\paren {\log N}^{B / 2 - 5} } } + \map \OO {\dfrac {N^2} {\paren {\log N}^{B/2} } }$

Now choose $B$ carefully.

$\blacksquare$


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Source of Name

This entry was named for Ivan Matveevich Vinogradov.


Historical Note

Vinogradov's Theorem: Corollary $2$ was conjectured by Edward Waring in $1770$ in Meditationes Algebraicae.

Ivan Matveevich Vinogradov proved what is now known as Vinogradov's theorem in $1937$, and as a result proved that $2$nd corollary: that every sufficiently large odd integer is the sum of three prime numbers.

This was considered an important step towards the resolution of Goldbach's Conjecture.


Sources

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