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The circle method represents a weighted additive count as an integral of products of exponential sums against a character. Near rationals with small denominator, major arcs produce a structured main term involving a singular series and an archimedean singular integral. Away from those rationals, minor arcs require cancellation bounds. The partition and approximation estimates depend on the problem; a formal integral alone is not a positivity proof.

Vinogradov's three-primes theorem states that every sufficiently large odd positive integer is a sum of three primes, with repetitions allowed. The parity restriction is a genuine local obstruction to the main asymptotic, not merely a technical convenience.

Let $n$ be large and define the prime-weighted exponential sumwhere $\Lambda$ is the Von Mangoldt function. Character orthogonality gives the Hardy-Littlewood circle method identityThe task is to extract a positive main term from rational neighborhoods and make the remaining integral smaller.

Take $Q=(\log n{)}^B$, with a fixed sufficiently large $B$, and use major arcs $|\theta -a/q|\le Q/n$ around reduced fractions with $q\le Q$. They are disjoint once $n$ is large relative to $Q^3$. The minor arcs are their complement. On a major arc put $\theta =a/q+\beta$. The Siegel–Walfisz theorem, followed by partial summation in each residue class, givesuniformly on these arcs after choosing the underlying logarithmic error exponent sufficiently large. The reduced residue-class sum is the Ramanujan sum $c_q(a)=\mu (q)$ when $(a,q)=1$. Prime powers dividing $q$ and the partial-summation factor $1+n|\beta |$ are absorbed in the stated error choice.

Cubing and integrating produces the singular series and singular integral:The last identity is Fourier inversion for the volume of the simplex $t_1+t_2+t_3=n$, $t_i\ge 0$; the upper bounds $t_i\le n$ do not restrict its interior. The tails and approximation errors are $o(n^2)$. Since $\mu (q{)}^3=\mu (q)$ on squarefree $q$, multiplicativity givesFor even $n$ the factor at $2$ vanishes. For odd $n$ it is $2$, andThe lower bound is uniform in odd $n$. The product converges because the sums of the local deviations converge. It describes how congruence restrictions alter the otherwise random prime-summand density.

For the minor arcs, a Type I/II prime-sum estimate provides the necessary cancellation. One useful form, obtainable from Vaughan's identity and bilinear estimates, iswhen $|\theta -a/q|\le q^{-2}$. Dirichlet's approximation theorem supplies a denominator $q\le n/Q$ with $|\theta -a/q|\le Q/(qn)$. On the minor arcs its denominator must exceed $Q$, so both denominator-dependent terms are at most $n/\sqrt{Q}$. Choosing $B$ large gives ${\sup }_{\mathfrak{m}}|S(\theta )|=o(n/\log n)$. The mechanism is to split $\Lambda$ into short-divisor Type I sums and bilinear Type II sums; geometric-sum bounds treat Type I, while Cauchy-Schwarz exposes correlations and rational spacing in Type II. The displayed estimate is the substantive analytic input, rather than a consequence of bounding each summand separately.

Meanwhile Parseval identity and the elementary prime-weight second moment giveThereforeCombining the two regions yields $R_{\Lambda }(n)=\frac{1}{2}\mathfrak{S}(n)n^2+o(n^2)$. Proper prime powers account for only $O(n^{3/2}(\log n{)}^4)$: there are $O(\sqrt{n}\log n)$ such choices for one coordinate, at most $n$ choices for another, and each weight is at most $\log n$. Hence they cannot account for the positive main term. The weighted count using primes alone is positive for every sufficiently large odd $n$, proving the asserted representation. The conclusion is the sufficiently-large three-prime theorem; no verification of every small odd integer is supplied by this argument.