Fixed-divisor shift of the Dickman main term

Let $X,y,d$ be natural numbers with $y\ge2$, $0<d\le X$, and $\log X\le5\log y$. Write $u(X,y)=\log X/\log y$ and let $\rho$ be the Dickman function. Then

This quantifies both the floor loss and the change of the Dickman parameter caused by a fixed divisor.