Pointwise variation of the natural theta weight

Let $2\le a\le b\le X$ be natural numbers, $a\le m<b$, and $\log X/\log a\le5$. Put $q_m=\lfloor X/m\rfloor$, $u_m=\log q_m/\log m$, and $J_m=I(q_m,m)$, where $I$ is the base-free fractional correction. Let $\Theta_X(m)$ be the natural Saias main term at $(q_m,m)$ divided by $\log m$, and let $S_X(m)=q_m/\log m-q_{m+1}/\log(m+1)$ be the selector cell ledger.

With $\rho$ the Dickman function,

This separates coefficient drift from the two globally summable smooth variations.