Common-range formula for a quotient drop and base increment

Let $1\le q_1\le q_0$ and $m\ge2$ be natural numbers with $\log q_0/\log m\le5$. 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 common cap permits variation estimates across changing natural quotients.