Tao Section 5: the logarithmic integral of the Type II estimate

For reals $0<b\le c$,

With $b=V$ and $c=x/U$ this is the integral that converts the pointwise bound on the dyadic Type II sums into the Type II estimate of the source's minor-arc theorem: the bilinear sum is written as $4\int_0^\infty F(W)\frac{dW}{W}$, the integrand is supported in $V\le W\le x/U$, and each of the resulting pieces carries a factor $\log W$. The right-hand side then reads

which is the shape in which the source records it.

The identity is the fundamental theorem of calculus for the primitive $\tfrac12\log^2W$ of $\log W/W$, together with $\log^2c-\log^2b=(\log c-\log b)(\log c+\log b)=\log\frac cb\log(cb)$.

Formalization Note The source's display writes the integral as $4\int_{V\le W\le x/U}\frac{dW}{W}$, without the factor $\log W$; the stated value $2\log\frac{x}{UV}\log\frac{Vx}{U}$ is the value of the integral with that factor, which is the one the argument uses, and is what is proved here. The integral is the interval integral with respect to Lebesgue measure.