Bilinear sum ($B(a,b)={\sum }_{m,n}{a}_m{b}_{n}K(m,n)$)

A bilinear sum has the formwhere $M,N$ are finite index sets and the coefficient sequences $a,b$ can be controlled separately. It is a bilinear form in these two coefficient lists. In analytic number theory, the function $K(m,n)$ often oscillates with the product $mn$, as in a Type II sum.

The Cauchy-Schwarz inequality eliminates one coefficient list and exposes correlations:Useful cancellation comes from small off-diagonal correlations, not from the bilinear notation alone. For example, take $m,n=0,\dots ,q-1$ and $K(m,n)=e^{2\pi imn/q}$. Orthogonality of roots of unity makes the inner correlation $q$ for $n=n^{\prime }$ and zero otherwise, giving $|B|\le \sqrt{q}\Vert a{\Vert }_2\Vert b{\Vert }_2$. The termwise triangle inequality only gives $q\Vert a{\Vert }_2\Vert b{\Vert }_2$. This explains why Fourier analysis and large sieve estimates are effective tools for bilinear sums.

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.

If $I_m$ is an integer interval inside $[1,N]$, discrete Fourier inversion gives the displayed identity. The exponential geometric sum bound gives a common envelope $H_N(\theta )\ll \min (N,\Vert \theta {\Vert }^{-1})$, whose integral is $O(\log (2N))$. At each frequency, $\widehat{1_{I_m}}(\theta )/H_N(\theta )$ is a bounded coefficient depending only on $m$, while $e(n\theta )$ depends only on $n$. A uniform bounded-coefficient Type II sum estimate therefore survives an $m$-dependent interval cutoff at a cost of one logarithm. This is particularly useful for the condition $mn\le N$.

Let $\alpha$ be a badly approximable number, and sum $B={\sum }_{D<d\le 2D}{\sum }_{W<w\le 2W,dw\le X}{a}_d{b}_{w}e(\alpha dw)$ with $|a_d|\le \tau (d)$ and $|b_w|\le 1$. The Cauchy-Schwarz inequality, divisor-square summatory bound and geometric sums give $|B{|}^2\ll D{\log }^{3}X(DW+W{\sum }_{h\le W}\min (D,\Vert h\alpha {\Vert }^{-1}))$. Separated rotations bound the last sum by $O_{\alpha }(W\log (2W))$, giving the displayed estimate. A hyperbolic cutoff remains valid because the allowed first-variable values after expansion form an interval.

A Type II sum has the form ${\sum }_{d\sim D}{\sum }_{m\sim M}{a}_d{b}_{m}f(dm)$ with two nontrivially weighted variables, both long enough for a Cauchy-Schwarz inequality and a large sieve estimate. The Vaughan identity supplies this bilinear structure for sums weighted by the Von Mangoldt function.

Suppose $|a_u|\le \tau (u)$, $|b_d|\le \log (2N)$, $u,d>X\ge 1$, $ud\le N$, and $|\theta -a/q|\le q^{-2}$ with $(a,q)=1$. A dyadic decomposition reduces $S=\sum a_u{b}_{d}e(\theta ud)$ to blocks $u\asymp U,d\asymp D$. The Cauchy-Schwarz inequality, divisor-square summatory bound and the reciprocal fractional-part sum near a rational giveThe condition $ud\le N$ makes the allowed $u$ values for each pair $d_1,d_2$ an interval, so the exponential geometric sum bound applies despite the hyperbolic cutoff. Since $UD\le N$ and $U,D\gg X$, take square roots and sum the $O({\log }^2(2N))$ blocks to get the displayed estimate.

A Type I sum in analytic number theory has the form ${\sum }_{d\le U}{a}_d{\sum }_{m\in I_d}f(dm)$, with a short weighted variable and a relatively long inner sum whose coefficients are simple. The Vaughan identity produces these sums when the Möbius function is truncated.

A dyadic decomposition partitions a positive range into intervals $(M,2M]$. It reduces a variable-length sum to $O(\log N)$ sums in which every variable has a fixed order of magnitude.