Type II sum
= Type II sum
A Type II sum has the form $\sum_{d\sim D}\sum_{m\sim M}a_db_mf(dm)$ 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>.