Step 1: Circle method setup. Write $R_3(N) = \int_0^1 S(\alpha)^3 e^{-2\pi iN\alpha}\,d\alpha$ where $S(\alpha) = \sum_{p \leq N} (\log p) e^{2\pi ip\alpha}$.

Step 2: Major arcs. For $\alpha$ near $a/q$ with $q \leq (\log N)^B$ and $|\alpha - a/q| \leq (\log N)^B/N$: by the Siegel-Walfisz theorem, $S(\alpha) \approx \frac{\mu(q)}{\phi(q)} v(\beta)$ where $\beta = \alpha - a/q$ and $v(\beta) = \sum_{n \leq N} e^{2\pi in\beta}$. The major arc integral evaluates to $\mathfrak{S}(N)\mathfrak{J}(N)$ where $\mathfrak{J}(N) = \int_\mathbb{R} v(\beta)^3 e^{-2\pi iN\beta}\,d\beta = \frac{N^2}{2} + O(N)$.

Step 3: Minor arcs (Vinogradov's key estimate). The critical innovation is the bound: for $\alpha$ on the minor arcs ($q > (\log N)^B$ or $|\alpha - a/q|$ large), $|S(\alpha)| \ll \frac{N}{(\log N)^A}$ for any $A > 0$.

This uses Vaughan's identity to decompose $S(\alpha) = S_1 + S_2 - S_3$ into Type I and Type II sums:

• Type I: $\sum_{m \leq U} a_m \sum_{n} e^{2\pi imn\alpha}$ (inner sum is geometric, bounded via $\|\alpha m\|$ estimates)
• Type II: $\sum_{m \sim M} a_m \sum_{n \sim N/M} b_n e^{2\pi imn\alpha}$ (bilinear, bounded via Cauchy-Schwarz and the large sieve)

With $U = N^{2/3}$, both types are $O(N/(\log N)^A)$ on the minor arcs.

Step 4: Combining. The minor arc contribution is $|\int_\mathfrak{m} S^3 e^{-2\pi iN\alpha}d\alpha| \leq \sup_\mathfrak{m}|S(\alpha)| \int_0^1 |S(\alpha)|^2 d\alpha \ll \frac{N}{(\log N)^A} \cdot N = \frac{N^2}{(\log N)^A}$, using Parseval: $\int|S|^2 d\alpha = \sum (\log p)^2 \sim N$.

For $A > 3$, this is $o(\mathfrak{S}(N)N^2/(\log N)^3)$, proving $R_3(N) > 0$ for large $N$. $\square$