Let $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function in analytic number theory, and there is a large literature on computing upper and lower bounds on $\Psi(x,y)$.
What are the best explicit upper and lower bounds known for $\Psi(x, y)$? The best explicit upper bound I have been able to find is $\Psi(x,x^{1/u}) \leq x \cdot e^{-u/2}$ when $x \geq 2^u$ (Tenenbaum, "Introduction to Analytic and Probabilistic Number Theory," Section III.5, Theorem 1), and the best explicit lower bound I am aware of is $\Psi(x, x^{1/u}) \geq x/(\log x)^u$ (see Lichtman and Pomerance, "Explicit Estimates for the Distribution of Numbers Free of Large Prime Factors," J. Number Theory, 2018). But in each case several more pages are devoted to better -- but ''non-explicit'' -- bounds (i.e., they are either asymptotic, involve unspecified constants, or involve functions that cannot easily be evaluated).
Can anyone point me to the best known explicit upper/lower bounds? If it helps, in my setting $x, y \gg 10^6$ and $x^{1/3} \geq y \geq x^{1/20}$.
Since $\Psi(x, x^{1/u})$ tends to $x \cdot \rho(u)$, where $\rho$ is the Dickman function, pointers to explicit upper/lower bounds on the Dickman function would also be helpful.