The special case $k=3$: van der Corput's theorem

Step 1 (weighted count via exponential sums). Write $S(\theta) = \sum_{n \le N} \Lambda(n) e(\theta n)$. The weighted count of 3-term progressions $p_1 + p_3 = 2p_2$ with all terms $\le N$ equals $\int_0^1 S(\theta)^2 S(-2\theta) \, d\theta$ by orthogonality of the exponential.

Step 2 (major arcs). Near rationals $\theta \approx a/q$ with $q$ small, $S(\theta)$ is well approximated using the prime number theorem in arithmetic progressions; summing these contributions gives the Hardy–Littlewood main term $\mathfrak{S}(N) \, N^2 / \log^3 N$, where the singular series $\mathfrak{S}(N)$ is a positive constant depending only on local (mod $q$) densities of primes.

Step 3 (minor arcs). Away from rationals with small denominator, Vinogradov's estimates bound $S(\theta)$ by $O(N (\log N)^{-A})$ for any fixed $A$, using cancellation in the sum over primes; integrating this bound over the minor arcs shows their total contribution is $o(N^2/\log^3 N)$ — negligible next to the major-arc main term.

Step 4 (conclusion). Since the major-arc main term $\mathfrak{S}(N)\,N^2/\log^3 N$ dominates the negligible minor-arc error, the weighted count of 3-term progressions grows like $N^2/\log^3 N \to \infty$, so there are infinitely many (and asymptotically many) 3-term arithmetic progressions of primes — a full 65 years before the general case was settled.