On integers whose friable component is large

Abstract

We estimate the number of integers up to $x$ whose $y$-friable component exceeds $z$. Uniformly for $\max\{2,\log z\}\leqslant y\leqslant z\leqslant x/2$, we determine the logarithm of this counting function up to an error $O(\log z/\log y)$. This sharpens an estimate arising from a problem in Ford's sieve methods lecture notes. The proof combines classical friable-number estimates with an elementary argument for the lower bound.