 Bilinear sum

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.