Vinogradov Type I–II method

This method uses cancellation in Type I sums and Type II sums to control a sum weighted by the Von Mangoldt function. The Vaughan identity separates the short-factor contributions from bilinear ones. Fourier separation of interval cutoffs permits a hyperbolic product cutoff. Ben Green's 2007 notes, Sections 2.3–2.4 give a version adapted to binary digit sums.

Extend $f:[1,N]\cap \mathbb{Z}\to \mathbb{C}$ by zero, and assume $|f|\le 1$. In one dyadic endpoint convention, Type I sums are $\delta$-small if ${\sum }_{M\le m<2M}|{\sum }_{n\in I_m}f(mn)|\le \delta N$ for every dyadic $M\le N^{1/100}$ and every choice of integer intervals $I_m\subset [K,2K)$, $MK\le N$. Type II sums are $\delta$-small if $|{\sum }_{m\in [M,2M)}{\sum }_{n\in [K,2K)}{a}_m{b}_{n}f(mn)|\le \delta N$ for all dyadic $M,K\in [N^{1/100},N^{99/100}]$ and coefficients $|a_m|,|b_n|\le 1$. Empty tails are handled by zero extension.