Total variation of the fractional correction along a natural hyperbola

Let $2\le a\le b\le X$ be natural numbers with $\log X/\log a\le5$, and set $q_m=\lfloor X/m\rfloor$. Write $K(q,m,t)=D_\rho((\log q-\log t)/\log m)/\log m$ for the scaled Dickman kernel, where $D_\rho$ is the source-defined Dickman derivative, and $I(q,m)=\int_1^{m^5}\{t\}t^{-2}K(q,m,t)\,dt$ for its base-free fractional correction.

Then

The bound controls the smooth fractional component in weighted hyperbola summation.