Vinogradov's Theorem

From ProofWiki
Jump to: navigation, search

Theorem

Let $\Lambda$ be the von Mangoldt function.

For $N \in \Z$, let:

$\displaystyle R \left({N}\right) = \sum_{n_1 + n_2 + n_3 \mathop = N} \Lambda \left({n_1}\right) \Lambda \left({n_2}\right) \Lambda \left({n_3}\right)$

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

Let $\mathcal S$ be the arithmetic function:

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

where:

$p$ ranges over the primes
$p \nmid N$ denotes that $p$ is not a divisor of $N$
$p \mathrel \backslash N$ denotes that $p$ is a divisor of $N$.


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

$R \left({N}\right) = \dfrac 1 2 \mathcal S \left({N}\right) N^2 + \mathcal O \left({\dfrac {N^2} {\left({\log N}\right)^A} }\right)$

where $\mathcal O$ denotes big-O notation.


Corollary 1

Let:

$\displaystyle r \left({N}\right) = \sum_{p_1 + p_2 + p_3 \mathop = N} 1$

where $p_1, p_2, p_3$ are prime.


Then for sufficiently large odd integers $N$:

$r \left({N}\right) = \mathcal S \left({N}\right) \dfrac {N^2} {2 \left({\log N}\right)^3} \left({1 + \mathcal O \left({\dfrac{\log \log N} {\log N}}\right)}\right)$

where $\mathcal O$ denotes big-O notation.


Corollary 2

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


Outline of Proof


Proof of Theorem

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

$e \left({\alpha}\right) := \exp \left({2 \pi i \alpha}\right)$

Let $B > 0$, and set $Q = \left({\log N}\right)^B$.

For $1 \le q \le Q, 0 \le a \le q$ such that $\operatorname{gcd} \left({a, q}\right) = 1$, let:

$\mathcal M \left({q, a}\right) := \left\{{\alpha \in \left[{0 \,.\,.\, 1}\right] : \left|{\alpha - \dfrac a q}\right| \le \dfrac Q N}\right\}$

Let:

$\displaystyle \mathcal M := \bigcup{1 \mathop \le q \mathop \le Q} \bigcup_{\substack{0 \mathop \le a \mathop \le q} {\operatorname {gcd} \left({a, q}\right) \mathop = 1}} \mathcal M \left({q, a}\right)$

be referred to as the major arcs.

Let:

$\mathcal m := \left[{0 \,.\,.\, 1}\right] \setminus \mathcal M$

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.


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

$\displaystyle R \left({N}\right) = \int_0^1 F \left({\alpha}\right)^3 e \left({-N \alpha}\right) \ d \alpha$



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

$\left[{0 \,.\,.\, 1}\right] = \mathcal m \cup \mathcal M$

we have:

$\displaystyle R \left({N}\right) = \int_{\mathcal m} F \left({\alpha}\right)^3 e \left({-\alpha N}\right) \ d \alpha + \int_{\mathcal M} F \left({\alpha}\right)^3 e \left({-\alpha N}\right) \ d \alpha$

We consider each of these integrals in turn.

Sum Over the Minor Arcs

For any $B > 0$:

$\displaystyle \int_\mathcal M F \left({\alpha}\right)^3 e \left({-\alpha N}\right) \, \mathrm d \alpha \ll \frac {N^2} {\left({\ln N}\right)^{B/2 - 5} }$

$\Box$

Sum Over the Major Arcs

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

Then:

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

where the implied constant depends only on $B$.

$\Box$

Putting these estimates together, we obtain:

$R \left({N}\right) = \dfrac {N^2} 2 \mathcal S \left({N}\right) + \mathcal O \left({\dfrac {N^2} {\left({\log N}\right)^{B / 2 - 5} } }\right) + \mathcal O \left({\dfrac {N^2} {\left({\log N}\right)^{B/2} } }\right)$

Now choose $B$ carefully.

$\blacksquare$



Source of Name

This entry was named for Ivan Matveevich Vinogradov.