Positive density of power-smooth integers ($\#\{n\le H:P^{+}(n)\le H^{\eta }\}{\gg }_{\eta }H$)

Fix $\eta =1/k$ with $k\ge 2$. Choose $k$ primes in $[H^{\eta -{\eta }^2},H^{\eta }]$, repetitions allowed, and an integer cofactor $m$ so that $m{p}_1\cdots p_k\le H$. The product is between $H^{1-\eta }$ and $H$, so $m\le H^{\eta }$ and all resulting integers are $H^{\eta }$-smooth. The Mertens theorem for reciprocal primes makes the reciprocal prime sum tend to $-\log (1-\eta )>0$. Since $\lfloor H/(p_1\cdots p_k)\rfloor \ge H/(2p_1\cdots p_k)$, there are ${\gg }_{\eta }H$ ordered representations. Each integer has only $O_{\eta }(1)$ prime factors in that interval, counted with multiplicity, so representation multiplicities are bounded by a constant depending on $k$. This proves the density bound. For arbitrary fixed positive $\eta$, choose a smaller reciprocal-integer exponent.

// Target: number-theory.bigb